PC-SIG: World of Education

home *** CD-ROM | disk | FTP | other *** search

/ PC-SIG: World of Education / PC-SiG's World of Education.iso / run / 3890 / hypergeo.doc < prev next >

Wrap

Text File | 1993-01-31 | 271.6 KB | 5,861 lines

HYPERGEO Version 2.1 A program for investigating the geometrical properties of four-dimensional hyperspace USER'S GUIDE AND PROGRAM REFERENCE MANUAL WareCraft P.O. Box 139 Bedford, MA 01730 USA HYPERGEO Version 2.1 Page ii COPYRIGHT NOTICE HYPERGEO is Copyright (c) 1993 by WareCraft. This document is Copyright (c) 1993 by WareCraft. ALL RIGHTS RESERVED. Contains licensed program materials of Metagraphics Software Corporation. Copyright (c) 1986-1989 Metagraphics Software Corporation, Scotts Valley, CA 95066 Self-extracting archives use LHA 2.13 Copyright (c) Yoshi, 1991. DISCLAIMER OF WARRANTIES AND LIMITATION OF LIABILITIES Due to the nature of computer programming it is impossible to guarantee the performance of this product. The author makes no warranty, expressed or implied, in regard to this program and its accompanying data files and documentation. In no event shall the author of this program be liable for incidental or consequential damages relating to or arising out of its use. HYPERGEO Version 2.1 Page iii LICENSE HYPERGEO is a copyrighted computer program. It is not public domain or free software. Copies of the program are made available on a trial basis to prospective users according to the distribution system known as "shareware". Persons acquiring a shareware copy of HYPERGEO are granted a limited license to use the program for an evaluation period not to exceed 30 days. At the end of that time they must either become registered users of HYPERGEO by paying the requisite registration fee or cease using the program. Recipients of the shareware copy of HYPERGEO may make duplicate copies of the program material for the express purpose of distribution to other prospective users, provided that: * Each copy contains all HYPERGEO program files, data files, and documentation files complete and unmodified; * No payment is charged for the shareware copies beyond normal charges for the cost of the diskette and its shipment and handling. Registered users of HYPERGEO are granted license to use the software without time limit on a single computer. A registered copy of the HYPERGEO program material may not be duplicated other than for back-up purposes. A registered copy of HYPERGEO may not be distributed to others. HYPERGEO Version 2.1 Page iv _______ ____|__ | (R) --| | |------------------- | ____|__ | Association of | | |_| Shareware |__| o | Professionals -----| | |--------------------- |___|___| MEMBER ASSOCIATION OF SHAREWARE PROFESSIONALS OMBUDSMAN POLICY This program is produced by a member of the Association of Shareware Professionals (ASP). ASP wants to make sure that the shareware principle works for you. If you are unable to resolve a shareware-related problem with an ASP member by contacting the member directly, ASP may be able to help. The ASP Ombudsman can help you resolve a dispute or problem with an ASP member, but does not provide technical support for members' products. Please write to the ASP Ombudsman at 545 Grover Road, Muskegon, MI 49442-9427 or send a CompuServe message via CompuServe Mail to ASP Ombudsman 70007,3536. HYPERGEO Version 2.1 Page v Table of Contents Section 1. Introduction.................................... 1-1 Section 2. From Hyperspace to the Video Screen............. 2-1 2.1 From Four Dimensions to Three................. 2-1 2.1.1 Projection.................................. 2-1 2.1.2 Intersection................................ 2-2 2.2 Direct Three-Dimensional Geometry Definition.. 2-2 2.3 From Three Dimensions to Two.................. 2-3 Section 3. The HYPERGEO Program............................ 3-1 3.1 General Structure............................. 3-2 3.2 The HYPERGEO Display Screen................... 3-4 3.2.1 Color....................................... 3-4 3.2.2 On-line Help................................ 3-7 3.2.3 Dialog Boxes................................ 3-8 3.2.3.1 Generic Text Dialog Boxes................. 3-9 3.2.3.2 File Selection Dialog Boxes............... 3-10 3.2.3.3 Color Selection Dialog Box................ 3-12 3.2.3.4 List Selection Dialog Box................. 3-13 3.2.3.5 Action Approval Dialog Box................ 3-13 3.3 Geometry Manipulations........................ 3-14 3.3.1 Rotations................................... 3-14 3.3.1.1 Orthogonal Orientations................... 3-18 3.3.1.2 The Locate Function for Rotations......... 3-19 3.3.2 Translation of the Intersecting Hyperplane.. 3-19 3.3.3 Moving the Viewpoint for 4-D Projections.... 3-20 3.3.4 Translation of Individual Vertexes.......... 3-22 3.3.4.1 4-D Image Considerations.................. 3-22 3.3.4.2 The Two User Interface States............. 3-23 3.3.4.3 Vertex Translations without a Mouse....... 3-24 3.4 View Control.................................. 3-25 3.4.1 The Five Display Modes...................... 3-25 3.4.2 Key Display Parameters...................... 3-26 3.4.3 Hyperface Highlighting...................... 3-27 3.4.4 W-coordinate Spectrum Banding............... 3-28 3.5 Journalling................................... 3-29 3.5.1 Recording a Journal File.................... 3-30 3.5.2 Replaying a Journal File.................... 3-31 3.6 PCX File Output............................... 3-34 HYPERGEO Version 2.1 Page vi Table of Contents (continued) Section 4. Inner Workings.................................. 4-1 4.1 Units......................................... 4-1 4.2 Verification of the Geometry Definition....... 4-1 4.3 Overlay Swapping.............................. 4-2 Section 5. Restrictions and Limitations.................... 5-1 5.1 Geometry Definition File...................... 5-1 5.2 Mathematical Precision........................ 5-2 Section 6. Supplied Geometry Definition Files.............. 6-1 Appendix A. Geometry Definition File........................ A-1 A.1 Text Format of the Geometry Definition File... A-1 A.2 The GEO2BIN Utility Program................... A-3 Appendix B. Configuration File.............................. B-1 Appendix C. Command Line Arguments.......................... C-1 Appendix D. Interactive Commands............................ D-1 D.1 Keyboard Commands and the Modifier Keys....... D-1 D.2 Mouse Selected Interactive Commands........... D-2 D.3 Interactive Parameter Specification........... D-5 D.4 HYPERGEO Interactive Keyboard Commands........ D-6 Appendix E. Running HYPERGEO under Microsoft Windows........ E-1 Appendix F. What's New in Version 2......................... F-1 Appendix G. Utility Programs for Geometry File Generation... G-1 HYPERGEO Version 2.1 Page 1-1 Section 1. Introduction "I tried turning the hypercube around, moving it away, bringing it up close, turning it around another way. Suddenly I could feel it! The hypercube had leaped into palpable reality, as I learned how to manipulate it, feeling in my fingertips the power to change what I saw and change it back again. The active control at the computer console created a union of kinesthetics and visual thinking which brought the hypercube up to the level of intuitive understanding." "The Mathematical Experience" Philip J. Davis & Reuben Hersh For reasons known only to the Great Cosmic Programmer, the universe in which we dwell possesses three spatial dimensions. Our senses of physical perception operate intuitively and naturally in these three dimensions. Indeed, it requires an act of intellectual imagination to consider the nature of alternative dimensional realities. With some contrivance we can picture the peculiar qualities of a world aware of only two dimensions. This is the substance of Edwin Abott Abott's famous fantasy "Flatland" (written from the viewpoint of Abott's two-dimensional narrator A. Square). We can make the mental adjustment required to conceive of a two-dimensional universe by imagining one of our three dimensions to be somehow squeezed down to zero extent and thus out of existence. It is a simple matter to draw pictures of two-dimensional objects; in fact, the nature of writing surfaces and writing implements is such that all our drawings are already limited to two dimensions (it is only the insistently three-dimensional nature of our sense of sight that causes us to "see" depth in our flat artwork). Perceptions in two dimensions are accessible because they already exist within our three-dimensional space and because we have at hand ready physical analogs in the form of flat surfaces of solid objects. In "Flatland", Abott makes his narrator aware of the existence of three-dimensional "Spaceland" although A. Square is unable to fully conceive of its nature. He further contemplates the analogous relationship between Spaceland and a still higher universe possessing four dimensions. In that scenario we, the inhabitants of our familiar three-dimensional reality, become the Flatlanders, constitutionally unable to perceive the totality of the encompassing higher realm. HYPERGEO Version 2.1 Page 1-2 When we try to imagine the shape and configuration of "hyperobjects" - that is, objects constructed in four dimensions - we are stopped short by the inability of our conceptual powers to find a place for the fourth dimension. For whatever reason, there simply is no way to twist or rearrange four separate orthogonal dimensions so they make spatial sense to us. Our best hope for grasping the essence of four-dimensional reality lies in somehow reducing the full complexity of hyperobjects to a simplified image that can be represented in three dimensions. Although the full four-dimensional "depth" of the hyperobject's geometry cannot be totally conveyed this way in a single image, it is possible by generating sequences of such images - each corresponding to a different orientation of the hyperobject in four-dimensional space - to begin to comprehend the shape of the fourth dimension. Naturally, computers and the capabilities of computer graphics are ideally suited for use as tools in generating such images, and it is well within the powers of the typical personal computer equipped with a standard display device to serve as a research platform for the investigation of hyperspace. HYPERGEO Version 2.1 Page 2-1 Section 2. From Hyperspace to the Video Screen When a computer is used to generate images derived from a four-dimensional hyperobject, two separate stages of geometric transformation must be performed. First, the four-dimensional object must be reduced to a particular three-dimensional image based on the object's orientation and position in hyperspace. Second, the resulting three-dimensional image must be represented on the flat, two-dimensional surface of the computer's display device. Because each of these transformations entails the loss of dimensional information, it is desirable to provide the computer user with as much control as possible over the manner in which the images are constructed and displayed. 2.1 From Four Dimensions to Three There are two basic approaches for generating a three-dimensional image from a four-dimensional object: projection and intersection. Each is highly analogous to the same technique applied to the reduction of a three-dimensional object to a flat, two-dimensional representation. 2.1.1 Projection Projection is the mathematically simpler of the two. It entails the construction of an imaginary "viewscreen" in hyperspace and the projection of each point in the hyperobject onto that viewscreen. The calculations are performed by considering the straight lines that run between each point in the hyperobject and a fixed four-dimensional viewpoint; the desired projection points are the intersections of these lines with the viewscreen. But because this is four-dimensional hyperspace, the viewscreen is itself a three-dimensional "hyperplane" (not just a simple two-dimensional plane), and the projection points all possess three dimensions. The positions of the viewpoint and viewscreen in hyperspace relative to the hyperobject determine the degree of "perspective" in the resulting three-dimensional image. (This process is entirely analogous to the projection of a 3-D object onto a flat 2-D viewscreen; in fact, just such a 3-D to 2-D projection is used subsequently to generate the final graphical output.) In a 4-D to 3-D projection, every point in the original hyperobject is always visible as a corresponding projected point in the 3-D image (although, of course, a point may from time to time happen to be hidden behind other points or lines). The HYPERGEO Version 2.1 Page 2-2 overall 3-D image that results is a network of connected lines corresponding to the vertexes and edges of the 4-D hyperobject; it is not itself a simple 3-D polyhedron. The complexity of this network relates directly to the four-dimensional structure of the hyperobject. 2.1.2 Intersection The second approach for reducing a four-dimensional hyperobject to a three-dimensional image involves calculating the intersection of the hyperobject with a three-dimensional hyperplane. This can be thought of as cutting a "slice" through the hyperobject and viewing the resulting cross-section. But again, since for such 4-D to 3-D intersections the cutting instrument is a hyperplane, the object that is determined by the cut is three-dimensional and forms a simple 3-D polyhedron - assuming that the original 4-D hyperobject is itself simple and is externally convex. (The 3-D to 2-D analogy of this involves making a planar cut through the 3-D geometry and viewing the flat polygonal outline of the cross-section.) In a 4-D to 3-D intersection, the only portion of the hyperobject that contributes to the 3-D image is that lying exactly on the intersecting hyperplane; all the rest is lost and invisible. For this reason, an intersection can result in very minimal 3-D polyhedra such as pyramids and box shapes, even from a complex hyperobject. In fact, if the intersecting hyperplane is positioned (in hyperspace) entirely outside the extents of the hyperobject, the resulting 3-D image will be empty. It is generally most revealing to examine the sequence of intersections that result when the cutting hyperplane is gradually moved through the hyperobject, starting at one face or edge and working across to the opposite. By taking a number of such sequences for various orientations of the hyperobject in four-space, the user can achieve a sense of its "extradimensional" structure. 2.2 Direct Three-Dimensional Geometry Definition The primary function of the HYPERGEO program is the display and examination of the properties of four-dimensional geometrical objects. The program's input is normally expected to be a geometry specification that provides four coordinate values for each vertex point. All of the preceding discussion has been in terms of this basic scenario. HYPERGEO Version 2.1 Page 2-3 However, the program will also accept geometry definitions of three-dimensional objects and will display them with all features available that relate to the 3-D level of processing. When used this way, the program bypasses the 4-D to 3-D image reduction steps described above, and uses the input geometry directly as the 3-D image definition. When the input geometry is three-dimensional, all of the program's commands that normally perform operations in four dimensions are deactivated while those commands and features that operate on the 3-D image and on its resultant 2-D representation are fully functional. 2.3 From Three Dimensions to Two All the options available in the HYPERGEO program for converting a three-dimensional image into a viewable two-dimensional picture on the computer's display are variations of the projection technique referred to above. Each 3-D point is projected onto a 2-D viewscreen by calculating where the line that runs between the 3-D point and a fixed 3-D viewpoint intersects the viewscreen. However, unlike the case for 4-D to 3-D projections, for 3-D to 2-D projections the viewscreen and viewpoint don't have to be invented; they correspond to the surface of the real display device and the eye of the (presumably real) viewer. Given this basic technique for projecting a 3-D image onto a flat screen, a great deal can be done to enhance the meaningfulness of the resulting 2-D picture. The following alternatives are available: * Perspective Projection A perspective projection is produced by performing a straightforward linear projection of the 3-D model onto the 2-D viewscreen using a realistic value for the distance between the viewpoint and the screen. It generates a wireframe representation that provides a measure of perspective which gives some sense of 3-D depth. This sense of depth perception is augmented by allowing the user to modify the apparent eye-to-screen distance, thus adjusting the amount of perspective in the picture. A perspective projection is the most basic form of 2-D image; all other imaging modes are enhanced variations of perspective projections. HYPERGEO Version 2.1 Page 2-4 * Stereoscopic Projection (Anaglyph) This uses the wireframe representation of a simple perspective projection, but two separate images are generated corresponding to the projections as perceived by the right and left eye (the calculation of the projections takes into account the distance between the eyes). When the two images are displayed in contrasting colors (typically red and blue or red and green), they can be viewed through filtering "3D" glasses which cause each eye to see only the single image generated for its particular viewpoint. The brain perceives the image with full three-dimensional depth. Although wearing the colored glasses is somewhat cumbersome, this method produces far and away the most vivid and revealing impressions of the structure of the 3-D object. (The general technique of generating dual projections that correspond to each eye's separate view is known as "stereoscopic" projection. The particular method of stereoscopic viewing that uses different colors for the two projections with matching colored filters for the eyes is called an "anaglyph".) * Hidden-line Projection This approach begins with the wire-frame representation generated by a normal perspective projection. Additional calculations are performed that prevent the display of any edge lying behind a face of the 3-D object. This causes the object to appear opaque. * Surface Relief Surface relief display mode combines the features of a hidden-line projection with a stereoscopic anaglyph. Using "3D" colored glasses, the viewer perceives the image with full three-dimensional depth, but because of the hidden-line processing, only those edges visible on the forward surface of the object are displayed. The wireframe representation appears opaque, presenting a perceptually vivid sense of the shape of its surface relief. HYPERGEO Version 2.1 Page 2-5 * Solid Projection A solid projection performs all the processing required by a hidden-line projection plus some additional calculations to determine the orientation of each visible face of the 3-D object. Rather than drawing the edge outlines of the object, the faces are filled in using either a density of fill pattern or a particular shade of color that corresponds to the obliqueness of the face as seen from a theoretical light source; faces that are most directly illuminated are filled the brightest. The result is a fairly realistic rendering of the 3-D object as solid. (Note: since hidden-line, surface relief, and solid projections require the 3-D object to be a simple polyhedron, they can only be used with 3-D images created by a 4-D intersection, not those created by 4-D projections.) HYPERGEO Version 2.1 Page 3-1 Section 3. The HYPERGEO Program HYPERGEO is a program written for the IBM-PC and compatible computers running the PC-DOS or MS-DOS operating system. It requires about 320K bytes of free memory plus whatever is needed for the data requirements of the particular geometry currently active in the program. For most ordinary objects (including the basic hypercube) memory requirements are quite modest, but they can increase considerably for very large and complex geometries. The program does not use expanded or extended memory for database allocations, so the 640K DOS barrier is the absolute limiting factor on the size of geometries that can be handled. The program will generate graphical output for EGA, VGA, or Hercules display devices. It does not support CGA graphics. For Hercules (and optionally for the others) the graphics are monochrome, and the stereoscopic anaglyph display modes are unavailable. The performance of HYPERGEO is identical on EGA and VGA systems with the exception of one area in which the VGA's higher performance is utilized: the enhanced VGA color capabilities are used to provide superior solid surface rendering. No non-standard, so-called "super-VGA" display modes are supported in this version. HYPERGEO performs some floating-point calculations, and the speed of the program will be enhanced somewhat by the presence of a math coprocessor chip such as an 80287 or 80387. However, the bulk of the program's mathematical operations are performed using an integer based, fixed-point format of data representation, so a system lacking a math coprocessor will perform almost as well. A coprocessor will be most useful on 80286 (or lower) PCs; systems with an 80386 CPU can perform 32-bit integer arithmetic which allows the program's special fixed-point calculations to be executed about as fast as coprocessor-accelerated floating-point. If the system has a math coprocessor, HYPERGEO will detect its presence automatically and make use of it. All features are functional without a coprocessor, but performance may be a bit slower (especially on 80286s). If a mouse pointing device is on the system, it can be used to invoke the interactive rotation and translation commands and to select other commands via the program's menu (see below). However, a mouse is not required since all commands can be executed through keyboard entry. Only the left and right mouse buttons are used, so a two-button mouse is as good as a three-button mouse. HYPERGEO Version 2.1 Page 3-2 The program will make use of a standard mouse driver if one has been installed. ("Drivers" are hardware device support programs that are run in background by DOS at system start up. They are typically "installed" by being specified in a DEVICE statement in the CONFIG.SYS file.) Most mice are supplied by their manufacturer with a standard mouse driver program, and the mouse's documentation will include directions for installing the driver. Other general-purpose mouse drivers - such as the one supplied with Microsoft Windows 3.x - are also standard and will enable proper use of the mouse by HYPERGEO. HYPERGEO also contains its own mouse support programming which permits the use of a Microsoft compatible mouse without any device driver. (Your system may be configured this way if it has a mouse which is used only by other applications which also contain their own internal device driver programming.) The mouse may be connected to either serial port, COM1 or COM2. 3.1 General Structure The HYPERGEO program accepts a description of the topology of a four-dimensional hyperobject. This description is contained in a "geometry definition file" which starts out as a plain text file that can be edited by the user. It contains coordinate information for each vertex point of the hyperobject plus connectivity information that indicates which vertexes are joined by edges. See Appendix A for a complete discussion of the format of this file. The text version of the geometry definition file is preprocessed into a more efficient binary form by the special utility program GEO2BIN. This program analyzes the topology and determines which edges form external faces and, in turn, which faces form external hyperfaces. GEO2BIN then writes the binary version of the geometry defintion into a new file which contains all the orginal vertex point and connectivity information plus the newly derived face and hyperface data. This binary file (which is not human-readable and cannot be edited) is what the main HYPERGEO program accepts as its primary input. See Appendix A for a fuller description of how to use the GEO2BIN utility. The program also accepts various parameters that control the manner in which the hyperobject is displayed. There are two ways in which these parameters are given to the program: through a special configuration file or as command-line arguments. (Note: There is not a complete correspondence between all possible configuration file parameters and all possible command-line arguments. Some parameters can be specified only in the configuration file and some only as command-line arguments.) See Appendix B for a full description of the format of the HYPERGEO Version 2.1 Page 3-3 configuration file and Appendix C for a description of the available command-line arguments. A section of configuration parameters can also be appended to the end of a geometry definition file following all topology information; the format is the same as in the primary configuration file. In effect, this acts as a secondary extension to the configuration file which will apply only to that particular geometry file. The order of precedence for conflicting parameters is as follows. Command-line arguments take the highest precedence and will always override a corresponding parameter in the configuration file or in the geometry definition file. Also, configuration parameters contained in a geometry definition file take precedence over corresponding parameters in the primary configuration file. Finally, every parameter has a pre-defined default value which will be used in the absence of any specification via the configuration file or the command line. The default values for all parameters are listed in the appendixes. To summarize the order of precedence for the various avenues of parameter specification, they are, from highest to lowest: * Command line arguments * Configuration parameters in current geometry definition file * Configuration parameters in primary configuration file * Program's internal default values It should be noted that no configuration file or command-line arguments are absolutely required. The program will operate without them using reasonable default values. Once all geometry and configuration information has been read in and processed, the HYPERGEO program creates a display of the image of the hyperobject. At this point, the user has access to a set of commands which facilitate the interactive manipulation of the hyperobject's display. These interactive commands are described throughout subsequent sections of this document and are summarized in Appendix D. HYPERGEO Version 2.1 Page 3-4 3.2 The HYPERGEO Display Screen The screen displayed by the HYPERGEO program consists of two areas: the primary graphics area where the image of the geometrical object being examined is displayed, and an optional menu and mouse control pad area. The user may select whether or not to display the menu area. If it is not displayed, the primary graphics area is the entire screen. When the menu area is displayed, it occupies the right edge of the screen, and the primary graphics area fills the entire remaining screen. The menu area lists a number of display parameters that may be modified interactively by the user. It also contains a panel of command selection boxes that may be activated by the mouse to perform various manipulations of the object being displayed. A mouse is not required for the HYPERGEO program; all mouse functions may also be invoked through keyboard commands. However, the mouse does provide a more naturally intuitive feeling of control over the manipulation of the geometry being studied. The mouse functions are only available when the menu area is displayed. (All keyboard commands are always available regardless of whether the menu area is displayed or not.) The HYPERGEO program automatically scales and centers the geometry display within the primary graphics area. The user may subsequently modify the scale and position of the display as desired. 3.2.1 Color For systems with color EGA or VGA displays, the user has control over the colors used to draw the various components which comprise the overall graphical output. The colors are specified using statements in the configuration file and may be set to any one of the sixteen standard EGA colors (listed in Appendix B). In addition, the color used to draw the actual edge lines of the geometrical object itself can be modified interactively from within the HYPERGEO program. In the absence of any user specification, HYPERGEO uses a set of default colors which are believed to provide an attractive and comfortably viewable appearance (although, of course, such matters are subject to the contentiousness of individual tastes). HYPERGEO Version 2.1 Page 3-5 There are eight separate components that together make up the total HYPERGEO display, and each can be assigned a different color: * Primary background This is the background color which covers the entire screen (including the optional menu area) before any graphics are drawn. It may be any color, but a dark shade such as black or the default dark gray is recommended for visibility. This is especially so when using the stereoscopic (anaglyph) display modes, which require a dark background to produce a good 3-D effect. (Black is probably the most vivid in this respect, but dark gray is a bit easier on the eyes for extended viewing.) Default: dark gray (Note: On many EGA displays the default "dark gray" tends to appear somewhat blue and not very dark; in this situation it is probably better to use black for the primary background. For VGA systems, HYPERGEO alters the definition of "dark gray" to a shade that is reliably both darker and grayer.) * Primary foreground This is the color used to draw the actual edge lines of the geometrical object as it is projected onto the computer's display screen. This color is also used for the transient data values displayed in the menu area and for most text appearing in the various dialog boxes. Naturally, the primary foreground color should give a good contrast with the primary background color. Default: white * Menu This color is used for drawing the outlines and fixed labels of the data items and the mouse control pad section in the menu area of the HYPERGEO program. It is also used for the display of the name of the geometry definition file which the user may select to have appear in the upper, left corner of the primary graphics area, and for the border color and background shading in dialog boxes. Default: yellow HYPERGEO Version 2.1 Page 3-6 * Geometry axis vectors As the user rotates the geometrical object in hyperspace, the HYPERGEO program keeps track of the orientation of each of the axes of the object's original coordinate system. A projection onto 3-space of the unit vector drawn from the origin along each axis may optionally be displayed as part of the primary geometry graphics. This assists in visualizing the displacement of the original orientation of the object in all the dimensions of hyperspace. Default: cyan * Message window background This is the background color for all prompting and message dialog box windows. Default: brown * Help window background This is the background color used in the on-line help windows. Default: light gray * Help window text This is the color used for the text in the on-line help windows. Default: black * Hyperface highlighting For 4-D geometries the user may select to highlight the portion of the visible graphics that belongs to one particular hyperface in the original hyperobject. This is done by using a different color to display all edge lines belonging to the highlighted hyperface. Default: light magenta HYPERGEO Version 2.1 Page 3-7 Color is also used in the surface rendering of the solid display mode, which may be used with images produced by 4-D intersections, but its use and configurability are complicated by the differing capabilities of EGA and VGA display devices. For EGA systems, the colors used for solid surface rendering are fixed by the program and are not modifiable by the user; these colors give an acceptable gradation of shades from light to dark and are not colors available in the sixteen standard EGA colors. For VGA, the program takes full advantage of the enhanced color capabilities of the display device, in particular its ability to generate over 256,000 discrete shades. HYPERGEO uses twelve variations - ranging from very light to very dark - of a single VGA color to produce the consistent shadings necessary for realistic solid surface rendering. (The program can also optionally make use of color "dithering" techniques - the pixel-by-pixel mixing of two different colors - to increase the effective number of discrete shades available for solid surface rendering to twenty-three.) The user can specify the base color (in terms of a VGA red/green/blue color definition) from which HYPERGEO will generate the twelve graduated shadings. For systems with a mouse, this base shade may be interactively modified using a color selection dialog box. For either EGA or VGA, the user may select not to use color for solid surface rendering, but to instead use variable pattern fill, which just uses the primary foreground color. Variable pattern fill is always used on monochrome systems. Several other HYPERGEO features make special use of color, but in general these colors do not accept any modification from the user. The stereoscopic anaglyph display modes use two shades of red and blue. However, these colors are fixed by the requirements of the "3D" viewing glasses and are not modifiable by the user. Also, the W-coordinate spectrum banding feature uses a spectrum of ten colors to indicate the W-coordinate value of edges displayed in 4-D projections and intersections. Again, however, these color values are determined by the program requirements and may not be respecified. 3.2.2 On-line Help On-line help is available whenever the primary graphics are visible and interactive editing is available within the HYPERGEO program. The help system is entered by typing F1. All help information exists in a set of screens or pages, each corresponding to a group of related commands. The commands are listed according to the keystroke that executes them along with a brief description of their function and usage. HYPERGEO Version 2.1 Page 3-8 There is one main, top-level help page which gives an overview of all HYPERGEO interactive commands and presents a menu listing the separate lower-level help pages. This main page is what first appears when the help system is invoked via the F1 key. The user then selects a particular page by typing its number. While in the help system, typing F1 always returns to the main help menu, and typing any valid help page number always brings up that individual page. To leave the help system and return to the HYPERGEO display type <Esc>; this restores the screen to its previous state. 3.2.3 Dialog Boxes Some of the interactive commands will prompt for additional user input (such as a numeric value, a color name, or a file name). This is done using an interactive window known as a "dialog box" which pops up in the middle of the primary graphics area. There are several different types of dialog box used by HYPERGEO depending on the context of the command. They can be divided into two major categories: those which request data to be entered or selected by the user, and those which merely display an informational message. All dialog boxes remain visible on the screen until they are actively dismissed by the user; this can be done by using the mouse to select the "Ok" button that appears at the bottom of the dialog box. For those types of dialog box that request user input, selecting "Ok" tells the program to accept the input as specified and to go ahead and execute the command. Dialog boxes that just display a message have only this single "Ok" button; it should be selected once the user has read and understood the box's contents. Dialog boxes that request input have a "Cancel" button in addition to the "Ok" button; selecting "Cancel" tells the program to disregard any input the user may have entered in the dialog box and to cancel the command and dismiss the dialog box's window. Note: For all mouse selections within HYPERGEO dialog boxes either the left or the right mouse button can be used; they are equivalent. Also, whenever a dialog box is active, the mouse cursor changes to a hand icon with a pointing finger. Typing <Enter> when a dialog box is active selects the "Ok" button, and typing <Esc> selects the "Cancel" button. These keyboard equivalents can be used to perform dialog box functions on systems without a mouse. HYPERGEO Version 2.1 Page 3-9 Dialog boxes that merely contain information are used whenever an error condition (such as invalid command input) must be reported. No user input is asked for; the window serves only to convey a message. Once it has been read, the user may dismiss the window by selecting the "Ok" button with the mouse (or by pressing <Enter> or <Esc>). One special instance of a message dialog box is the HYPERGEO Information Window. This window contains some basic data describing the geometry and topology of the currently active object, and it also contains the HYPERGEO copyright statement. By default, the Information Window is displayed automatically upon the initial start-up of the program and every time a new geometry definition is read in (although the program may be configured to bypass this). An interactive command exists to display the Information Window at any time during normal operation of the HYPERGEO program. The Information Window is a normal message dialog box and may be dismissed by selecting the "Ok" button with the mouse (or by pressing <Enter> or <Esc>). Dialog boxes that request user input are one of the following special types: * Generic text * File selection * Color selection * List selection * Action approval (yes/no) 3.2.3.1 Generic Text Dialog Boxes A number of HYPERGEO's interactive commands require the user to type in a string of text. Typically, this type of entry is used to specify a numeric value (or possibly a sequence of two or more values) to be assigned to a display parameter. The text dialog box contains a title bar identifying its function and, below it, an area in which appears the text entered by the user. A special cursor symbol is visible in this type-in area indicating the point at which the next entered character will appear. The arrow keys may be used to move the cursor, and the standard line-editing keys such as <Delete>, <Backspace>, <Home>, and <End> are also functional. The special sequence Ctrl-U can be used to completely clear the contents of the type-in area and move the cursor to the first character position. HYPERGEO Version 2.1 Page 3-10 The user should type in the appropriate value which will appear in the data input area, and should terminate the entry by selecting the "Ok" button with the mouse (or by pressing <Enter>); this indicates that the entry is complete and correct and initiates the particular command. Selecting "Cancel" (or pressing <Esc>) at any time while a text dialog box is displayed will dismiss the dialog box and cancel the command. 3.2.3.2 File Selection Dialog Boxes HYPERGEO command functions that require the specification of a file name make use of the special file selection dialog box. There are currently two such commands: the New Geometry Definition File command (keystroke 'F'), and the Journal File Replay command (keystroke 'J'). The file selection dialog box has all of the features of a generic text dialog box plus a large sub-window in which a list of available file names is displayed, sorted in alphanumeric order. The sub-window has a scroll-bar along its right edge which can be used to move the list of file names up and down in the sub-window if the list is too long to be visible all at once. The user chooses a particular file name from the list by selecting its name in the sub-window with the mouse. When a file selection dialog box first pops-up, the text area contains a wild-card specification based on the default extension for the type of file required by the particular command. For example, for the Journal File Replay command, the wild-card *.JNL is used. All files that satisfy the wild-card specification will be listed in the sub-window, with no individual file name highlighted as being selected. Until a particular file is selected, the dialog box's "Ok" button is blanked out. Once a file name is selected, its name will appear in the text type-in area which is just above the list sub-window. The "Ok" button is then active and may be selected with the mouse thus confirming the name appearing in the text window as the file to be used by the command. (By double-clicking on a file name, the "Ok" button is immediately selected, confirming the file name and causing the HYPERGEO command to be performed on that file.) The text window in a file selection dialog box may also be used like a generic text input box, and a file name may be typed into it directly. To do this, it is first necessary to activate the type-in cursor by positioning the mouse in the type-in area and clicking the left or right mouse button. HYPERGEO Version 2.1 Page 3-11 It is also possible to enter a new wild-card specification by typing it into the text type-in area. This feature can be used to change the directory in which the command looks for its file (the default is always the current directory in which HYPERGEO was executed). For example, with the Journal File Replay command, the initial default wild-card is *.JNL; this will list all files with extension .JNL in the current directory. By positioning the cursor to the left of the '*' character and typing in its full DOS path specification preceding the *.JNL wild-card, the user can access a list of all .JNL files in another alternative directory. Within a file selection dialog box the <Enter> and <Esc> keys have dual uses depending on whether or not there is an active type-in cursor. When the type-in cursor is active and visible in the text area, the <Enter> key terminates the type-in and makes it the current selected item. If it is a wild-card specification, the contents of the main list sub-window are changed to reflect its expansion. When the type-in cursor is not active, the <Enter> key has its normal dialog box function of selecting the "Ok" button. Similarly, for the <Esc> key, when there is no active type-in cursor, its function is the normal dialog box selection of the "Cancel" button. When there is a type-in cursor visible in the text area, <Esc> causes all typed in changes to the text in that area to be discarded; the text area contents revert back to what they were before the type-in cursor was activated, and the cursor is deactivated. The scroll-bar has various active areas which can be selected with the mouse to adjust the position of the list of file names in the main sub-window. Selecting the up (or down) button at the top (or bottom) of the scroll bar causes the list to move up (or down) one line. Selecting the slider button and moving it up or down while keeping the mouse button pressed scrolls the list dynamically in step with the motion of the slider. Pressing a mouse button while the mouse cursor is in the scroll-bar slide area above or below the slider button scrolls the list up or down by an amount equal to the number of lines visible in the list sub-window. Most of the file selection dialog box functions have keyboard equivalents, and this enables the use of the dialog box on systems without a mouse. The type-in cursor can be activated in the text area by typing <Tab>. The up and down arrow keys perform the single-line up and down scrolling functions, and Page Up and Page Down perform multi-line scrolling. <Home> and <End> scroll the list all the way to the top or bottom respectively. Typing an alphanumeric character scrolls the file list so that those names beginning with the typed character appear at the top of the main sub-window. To select a particular file, scroll the list of file names until the desired one is at the top of the HYPERGEO Version 2.1 Page 3-12 list sub-window and type <Space>. And as with all dialog boxes, <Enter> corresponds to the "Ok" button (assuming the type-in cursor is not active), and <Esc> selects "Cancel" (again, assuming no type-in cursor). 3.2.3.3 Color Selection Dialog Box The color selection dialog box is used by the interactive color command when invoked to modify the shades used for solid surface rendering of an object. It permits the precise specification of the base color to be used for generating the solid fill shades through the setting of its constituent red, green, and blue components. Since only VGA displays have the capacity to display the full range of available colors, the color selection dialog box is only available on VGA-equipped systems. In addition, a mouse must be present to use this type of dialog box (all of the other types of dialog box used in HYPERGEO may be operated with keyboard equivalents to the mouse). If a system with VGA graphics has no mouse, a generic text dialog box will appear requesting the specification of the new color via the direct entry of three numeric values which are the respective intensities of the color's red, green, and blue components. The color selection dialog box has a main sub-window in which three "thermometer" controls appear, one for each of the three primary colors. At the base of each control is an active button area which is divided into two halves, one marked '+' and the other '-'. Selecting the '+' area with the mouse increases the intensity value of that control's color, and selecting '-' decreases it. The indicator of each thermometer control moves to show the current value of its color as it is adjusted between zero and the maximum. Above the control sub-window is a color panel area which displays the actual color corresponding to the composite of the red, green and blue components. As the controls are manipulated to change the intensities of the three primaries, the color visible in the panel continuously changes to show the resulting composite. Once the color has been adjusted to the desired shade, it may be confirmed by selecting the dialog box's "Ok" button (or typing <Enter>). This executes the interactive color command and modifies the display palette used for solid surface rendering based on the calculated shades derived from the new base color. Alternatively, by selecting "Cancel" (or typing <Esc>), the user can choose not to modify the current palette of solid fill shades. HYPERGEO Version 2.1 Page 3-13 3.2.3.4 List Selection Dialog Box In a number of interactive commands available in HYPERGEO the user needs to select one from a list of several possible alternatives. For example, the interactive command to change the display mode can potentially choose any one of five possibilities: Perspective, Stereoscopic Anaglyph, Hidden-line, Surface Relief, and Solid Surface Rendering. Commands such as this one that present a choice from a fixed set of options make use of the list selection dialog box. A list selection dialog box contains a "ballot" sub-window in which each of the selectable options appears on a single line. Each option is identified by a key word or two that appears next to an active selection button. A particular option is chosen by selecting its button with the mouse. When an option is selected, its button area is marked with an 'X'. When the dialog box first pops up, the option corresponding to the current state of HYPERGEO is marked as selected. Depending on the context of the particular command and possible interdependencies among elements of the HYPERGEO environment, not all options may be available at all times. Those options that are invalid during a particular command invocation are grayed out in the ballot sub-window of the dialog box, and their buttons are inactive for mouse selection. (As an example of this, not all five of the display modes listed above are available on monochrome systems or when a four-dimensional hyperobject is being viewed using a 4-D to 3-D projection.) Once a desired option has been selected in the dialog box, it may be confirmed by selecting "Ok" with the mouse (or typing <Enter>). Selecting "Cancel" (or typing <Esc>) leaves the HYPERGEO parameter or mode unchanged. Keyboard equivalents are available that enable the selection of list options without the use of a mouse. The up and down arrow keys move the selection up or down one line in the ballot area. <Home> and <End> are used to select the option at either the top or the bottom of the list. 3.2.3.5 Action Approval Dialog Box This type of dialog box doesn't really request any input from the user other than choosing between the "Ok" and the "Cancel" buttons. There is no text type-in or other interactive sub-window area. The title bar of the dialog box describes an action which the HYPERGEO program will tentatively take, and the user should select either the "Ok" button to accept the action or the "Cancel" button to cancel it. The prime example of the application of this type of dialog box is the Quit command (keystroke 'Q') which does not actually quit HYPERGEO and go back to DOS unless the user confirms it. HYPERGEO Version 2.1 Page 3-14 As always, typing <Enter> is equivalent to selecting "Ok", and typing <Esc> is equivalent to selecting "Cancel". 3.3 Geometry Manipulations Central to the operation of the HYPERGEO program is the ability to interact with the geometry as it is being displayed and to have the picture on the screen respond to the user's directions. It is just this ability to manually manipulate the hyperobject in all four dimensions of hyperspace that creates a sense of "hands-on" control and establishes an intuitive feel for the extradimensional shape of the hyperobject. In light of the two-stage process required to capture an image of a four-dimensional geometrical object on a two-dimensional computer display screen (first, 4-D to 3-D reduction, and second, 3-D to 2-D projection), HYPERGEO provides interactive functions to manipulate the geometry both in its original hyperspace configuration (in 4-space) and in its reduced three-dimensional representation (in 3-space). A word or two is needed here on the subject of coordinate systems. The HYPERGEO program uses the letters X, Y, Z, and W to label the four coordinates in hyperspace. Wherever ordering is significant (as, for example, in the specification of coordinates within a geometry definition file), the order is (X,Y,Z,W). As is conventional with computer graphics, three-dimensional coordinate systems are "left-handed"; thus, the positive Z-axis points away from the viewer and into the screen. 3.3.1 Rotations The primary method of examining an object is to turn it about to view it from many different angles. In this manner, it becomes possible to discover the overall nature of something which - for whatever reason - is not immediately accessible in its entirety to our perceptions. In the HYPERGEO program, rotations in all possible directions are available through interactive commands, and it is possible to position the geometry in whatever orientation is desired. Rotations can be performed either on the 4-D geometry before it is reduced to a 3-D image, or on the 3-D geometry of the reduced image itself. The effect of the two classes of rotation is different, not only because of how they are handled by the HYPERGEO program, but also because of fundamental differences between the structure of 3-space and that of 4-space. HYPERGEO Version 2.1 Page 3-15 In our familiar 3-space, rotations are performed about an "axis" which is a straight line in space. The rotation leaves all points on the axis unchanged while all other points in 3-space move in circles that are centered on the axis and that lie in a plane perpendicular to the axis. However, in 4-space, there are an infinite number of lines that pass through a given point on a plane and that are perpendicular to the plane. Thus when a hyperobject is rotated so that the points in a particular plane move in circles about a fixed point on the plane, it is not correct to say that the rotation takes place around an axis. It is most meaningful to our three-dimensional minds to think of 4-D rotations as being "on" or "of" a particular plane rather than being "around" anything. (Actually, rotations in hyperspace take place around a plane in the sense that for any given direction of hyperspatial rotation there is a plane on which no points move, namely, the plane determined by any two distinct lines perpendicular to the rotating plane and passing through the rotational centerpoint on it. This is virtually impossible to picture for our 3-D brains, so it is simpler to refer to such rotations in terms of the plane which is being rotated.) Any particular spatial orientation can be reached by performing an appropriate sequence of rotations of the planes determined by pair-wise combinations of the space's coordinate lines. In our 3-space there are three such planes: the XY-plane, the XZ-plane, and the YZ-plane. In hyperspace, which has four orthogonal coordinate axes, there are six planes that form the basis for all rotations: XY, XZ, XW, YZ, YW, and ZW. (In 3-space the rotations are normally referred to by the name of the axis around which the rotation occurs rather than the plane of the rotation.) HYPERGEO has interactive commands for performing rotations of all nine planes. For all HYPERGEO rotations, the coordinate system's origin is always the centerpoint; thus the origin is never moved by any rotation. (Note: the configuration file accepts a parameter, ORIGIN, for repositioning the origin within the original geometry; this in turn has the effect of locating the center of all rotations at any desired point in hyperspace.) All rotations in HYPERGEO are in incremental steps of the current Turn Angle. The value of the Turn Angle can be set by the user in the configuration file, and it can also be modified interactively at any time. Its current setting in degrees is displayed in the menu area. The Turn Angle can be set to a large number of degrees (say, 30 or 45) for rapid rotations of the object, and then reduced to a smaller value for fine positioning. Interactive commands exist for rotating the object one incremental step, through either a positive or negative Turn Angle, within any one of the three 3-D and six 4-D base rotational planes. Also, there are commands for automating any one of the possible modes of rotation; in these automated modes HYPERGEO continues to perform incremental rotations in the specified direction as rapidly as the computer's processing speed will allow. HYPERGEO Version 2.1 Page 3-16 The HYPERGEO program provides two features to assist the user in keeping track of the orientation of the active object in hyperspace as it undergoes various rotations. First, there is the ability to display the projected image of all four unit vectors emanating from the origin along the four hyperspace coordinate axes. This provides good visual feedback of how the object is turning in hyperspace. Second, the menu area contains data fields that display the angular displacement of each coordinate axis from its initial, unrotated position. These displacement angles can vary from 0 degrees (when the axis is coincident with its initial alignment) to 180 degrees (indicating that it has been rotated to point in the completely opposite direction). It is important to understand how four-dimensional rotations of the hyperobject in hyperspace differ from three-dimensional rotations of the reduced image in normal 3-space. The critical point is that rotations in 4-space affect the shape and structure of the reduced 3-D image and do not necessarily produce directly equivalent rotations of that 3-D image. Rotations in 3-space, on the other hand, are performed after the 4-D to 3-D reduction, and they always directly rotate the 3-D image; they never alter its shape. To clarify this, consider what is going on as the hyperobject is turned in hyperspace. If the program is using 4-D to 3-D intersection mode, the slicing hyperplane will pass through varying cross-sections of the object, and these cross-sections will be three-dimensional polyhedra of varying topologies. The 3-D polyhedra themselves will not necessarily rotate. For the projection mode of 4-D to 3-D reduction, rotations of the hyperobject will have the effect of altering the pattern of the network of connected edges that is projected into 3-space, but the motion of the network will not be any simple three-dimensional rotation. This distinction - that 4-D rotations alter the shape and structure of the reduced 3-D image, while 3-D rotations rotate the image - while generally true, is, alas, not the whole story. Four-dimensional rotations behave differently depending on whether or not the plane of the rotation contains the W-axis. This is due to the fact that in HYPERGEO the W-dimension is the one in which 4-D to 3-D image reductions take place: 4-D to 3-D intersections always occur along a hyperplane that is defined by a constant W value, and 4-D to 3-D projections always are performed onto a hyperplane that is similarly defined by a constant W value. For this reason, rotations that occur on planes that do not contain the W-axis do not alter the shape of the resulting 3-D image (because they do not modify the W-coordinate of any points in the hyperobject). Instead, they behave like 3-D rotations of the corresponding plane (for example, a 4-D rotation of the XY-plane has the same effect as a HYPERGEO Version 2.1 Page 3-17 3-D rotation around the Z-axis). The net outcome of all this is that the 4-D rotations of the XY, XZ, and YZ-planes look like simple 3-D rotations and have the same effect on the appearance of the object's display in HYPERGEO as 3-D rotations around the Z, Y, and X-axes, respectively. The other three 4-D rotations (that is, of the XW, YW, and ZW-planes) do alter the 3-D image's shape and do not resemble any simple 3-D rotations. One final point is needed to complete this discussion of rotations. Although 4-D rotations of the XY, XZ, and YZ-planes look the same as 3-D rotations around the Z, Y, and X-axes, they are not treated the same by the HYPERGEO program. The 4-D rotations permanently alter the orientation of the hyperobject in 4-space as that orientation is calculated and remembered by the program; they are cumulative in the sense that each subsequent 4-D rotation takes as its initial starting point the orientation arrived at by the previous 4-D rotation. 3-D rotations, on the other hand, do not permanently modify the four-dimensional orientation of the hyperobject; rather, they serve as temporary repositionings of the current 3-D image in 3-space. For this reason, 3-D rotations are most useful as a means of examining the shape of the current 3-D image expressly without altering the 4-D orientation of the hyperobject that produced that image. A 4-D rotation performed following a sequence of 3-D rotations reverts back to the most recent 4-D orientation for its starting position since that is the permanent orientation remembered by HYPERGEO. The transient 3-D rotations of the reduced image are lost and forgotten. (It is worth noting that although the 4-D rotations may look like 3-D rotations, they involve the additional computational effort of updating the permanent 4-D database and thus are noticeably slower.) This functional distinction between 4-D and 3-D rotations is reflected in the way HYPERGEO reports the hyperobject's orientation. In particular, the axis displacement angles (displayed in the menu area) and the axis vectors (displayed in the primary graphics area) are only affected by 4-D rotations. During temporary, 3-D rotations, they remain unchanged and continue to indicate the permanent 4-D orientation from which the next 4-D rotation will proceed. (Note that the above comment does not apply when the input geometry definition is a three-dimensional object rather than a 4-D hyperobject. For three-dimensional geometries the program treats all 3-D rotations as permanent modifications to the object's database; the displayed axis angle displacements and the unit axis vectors are changed upon 3-D rotations.) In addition to the commands which perform incremental rotations in any of the possible directions, HYPERGEO also has two methods for placing the current geometry directly in a particular orientation. These are discussed in the following two sections. HYPERGEO Version 2.1 Page 3-18 3.3.1.1 Orthogonal Orientations An orthogonal orientation is one in which all of the coordinate axes have been rotated an integral multiple of 90 degrees from their original positions. Such orientations permit viewing the object in a direction which is exactly perpendicular to one of the planes formed by a pair of coordinate axes. The HYPERGEO program provides interactive commands for generating all of the unique orthogonal orientations that are possible for both 3-D and 4-D geometries. In three-dimensional space there are three distinct orthogonal orientations; these correspond to viewing directions from the front, side, and top. Each of the three orientations is directed perpendicularly to one of the three 3-D coordinate axis planes. In four-dimensional space things are, of course, more complicated. Recall that for 4-D there are a total of six different pairs of coordinate axes that each form a distinct plane (these are the six planes about which the various incremental 4-D rotations are performed). Thus, there are at least six different orthogonal orientations, each derived from a view that is perpendicular onto one of the six coordinate planes. However, this is not the entire story. In four-space for each orientation perpendicular to a coordinate plane, the plane determined by the other two coordinate axes can rotate independently. In fact, it can be rotated so that its two coordinate axes are themselves positioned orthogonally along the two remaining coordinate directions in two different ways. This additional degree of freedom means that there are two different orientations based on each of the six possible base coordinate planes. This gives a total of twelve distinct orthogonal orientations for four-dimensional geometries. For geometries that represent regular polyhedra that are symmetrical about all rotations, all orthogonal orientations will produce an identical view. This is true for all of the regular 3-D polyhedra and 4-D polytopes that are provided with HYPERGEO. However, for shapes that possess asymmetries, the ability to view the object from all of the possible orthogonal directions provides an excellent way to see how the structure is arranged throughout all dimensions. For 4-D geometries, it is particularly interesting to examine orientations which position each of the four axes in turn in the direction of the original W-coordinate axis. Because the mechanisms for performing 4-D to 3-D image reductions operate along the W-axis, using the orthogonal orientations this way enables the resulting projections and intersections to reveal all of the hypergeometrical structure of the object. HYPERGEO Version 2.1 Page 3-19 3.3.1.2 The Locate Function for Rotations An interactive command permits specifying the orientation of the current object in terms of the three (for 3-D) or four (for 4-D) axis angles. These are the spatial angular displacements of each of the coordinate axes from their original unrotated positions. (The axis angles are displayed in the HYPERGEO menu area under the label AXIS ANG.) This command effectively permits placing the object at any precise orientation that is physically possible. For three-dimensional geometries, the user can specify any three angles that form a possible 3-D orientation. Each set of three axis angles determines a unique orientation in three-space. For four-dimensional objects the command operates in a similar manner, but there are some limitations. First, at least one of the specified angles must be zero. And second, most sets of axis angles do not correspond to a single orientation, but rather have multiple solutions. (This is due to the fact that the axis angles indicate only the magnitude of the displacement from the original axis and not its direction; thus the problem is mathematically underdetermined.) When this is the case, the program arbitrarily selects one particular orientation from among all of the valid possibilities. 3.3.2 Translation of the Intersecting Hyperplane For the intersection mode of 4-D to 3-D reductions, the program calculates the three-dimensional intersection of the hyperobject with a hyperplane that is defined by the equation, w = k, that is, having a constant W coordinate value. (In four dimensions, a linear equation of the four coordinate variables, Ax + By + Cz + Dw = E, defines a hyperplane. A hyperplane can perhaps best be envisioned as a normal three-dimensional space that is situated in hyperspace and has zero extent into the fourth dimension. Our universe at any given instant can be considered a hyperplane in four-dimensional space-time.) The shape of this intersection can be varied by rotating the hyperobject; this approach has been discussed in the above section on rotations. However, the intersection can also be modified by moving the hyperplane through hyperspace, in particular, by moving it along the W-axis by altering the value for 'k' in the hyperplane's equation. The HYPERGEO program provides interactive commands for performing such a translation of the intersecting hyperplane. (It should be emphasized that the position of the intersecting hyperplane only has an effect in the intersection mode of 4-D geometry reduction; it is irrelevant when using projection mode.) HYPERGEO Version 2.1 Page 3-20 Two parameters relate to the interactive translation of the intersecting hyperplane: the current W value of the hyperplane's equation, and the amount by which that value will be incremented (or decremented) for each translation of the hyperplane. The first of these values is displayed in the HYPERGEO menu area as the W-INTER parameter, and the second is referred to as the delta-W value. An interactive command exists that increments the W-INTER value by the delta-W value and updates the HYPERGEO display accordingly. Another command serves to decrement the W value and thus move the intersecting hyperplane in the reverse direction. In addition, both parameters may be explicitly set to any desired numeric value using other interactive functions. The initial values of the W-INTER and delta-W parameters may also be set in the HYPERGEO configuration file. If W-INTER is not specified by the user in the configuration file, its default value is calculated as the average of the minimum and maximum W values for the active geometrical object thus centering the intersecting hyperplane within the geometry. Translation of the intersecting hyperplane is most revealing when it proceeds from one edge or face of the hyperobject, through its entire body, and on to the opposite side. A feature exists which automates just such a traversal of the hyperplane. It continuously increments the W value until the body of the hyperobject is exited. The direction of translation is then reversed and the W value is continuously decremented until the opposite side of the hyperobject is reached. The direction of translation continues to be reversed at each extreme so that the hyperplane passes back and forth (in hyperspace) through the entire body of the hyperobject. Performing such passes upon varying orientations of the hyperobject in hyperspace provides an excellent method for acquiring a sense of its four-dimensional shape. (Note: the effective speed of the hyperplane's motion along the W-axis can be adjusted by setting the delta-W value appropriately.) 3.3.3 Moving the Viewpoint for 4-D Projections As described in a previous section, for the second primary method of reducing a 4-D hyperobject to a 3-D image, namely, a 4-D to 3-D projection, a theoretical viewscreen is set up (mathematically) in hyperspace and the hyperobject is projected onto it as seen from a theoretical viewpoint. (Recall that because the "viewscreen" is actually a three-dimensional hyperplane, the resulting image is also three-dimensional.) To provide enhanced control over this projection, an interactive command exists to modify the position of the theoretical viewpoint. This allows the user to adjust through a wide range the level of perceived perspective evident in the resulting 3-D image. HYPERGEO Version 2.1 Page 3-21 It should be noted that the position of the projection viewpoint only has meaning for images that are generated as 4-D to 3-D projections; it is irrelevant for 4-D to 3-D intersection images. The nature of the projection calculations restricts the viewpoint to the positive W-axis. This is due to the way in which the hyperplane which is the theoretical viewscreen is located at a fixed position and orientation in hyperspace. In particular, the viewscreen hyperplane is defined by the equation w = 0.5R where 'R' is the "radius" of the hyperobject. This places the viewscreen parallel to the XYZ-hyperplane and intersecting the positive W-axis at a distance 0.5R from the origin. The user can specify the projection viewpoint to be located anywhere along the positive W-axis within a large range of coordinate values. The program does, however, automatically prevent the viewpoint distance from assuming a value that is either too small or too large. This is done to prevent possible numeric errors in the calculation of the reduced image that uses the viewpoint coordinate. When the viewpoint is positioned towards the low end of its range, the degree of perspective in the calculated 3-D image is greatest. The perspective becomes less as the viewpoint moves away from the hyperobject and from the viewscreen along the W-axis. For the largest values of this distance, the apparent perspective is minimal, and the image closely approximates that of a pure orthographic projection. The position of the 4-D projection viewpoint is displayed as a parameter in the HYPERGEO menu area under the label W-PROJ. It appears as a numeric value in the units of the geometry definition. This value is the W-coordinate of the viewpoint. Interactive commands exist which allow incrementally increasing or decreasing the W-PROJ parameter by 10% of its current value. Additional commands set the W-PROJ parameter directly to either its maximum or minimum permissable value; this has the effect of selecting either orthographic or perspective projection. The parameter can also be set interactively to any specific intermediate value within its range, and it may be assigned a new default value in the HYPERGEO configuration file. (The program's own default value for W-PROJ is two times the radius of the hyperobject; this is close to the minimal permissable value and creates 4-D to 3-D projection images with a considerable degree of perspective.) HYPERGEO Version 2.1 Page 3-22 3.3.4 Translation of Individual Vertexes As described above, rotating a hyperobject is the primary means of examining its structure in HYPERGEO, and the user interface presented upon entry to the program provides access to the various commands for performing such rotations. However, an alternative state of the user interface is available which presents commands that can be invoked to perform the temporary translation of a selected vertex of the hyperobject and to observe how the translation of an object's vertex in hyperspace affects the shape and structure of the corresponding 3-D image. The selected vertex can be moved in either direction along any one of the four 4-D coordinates. An important point needs to be made concerning the non-permanent nature of vertex translations. HYPERGEO maintains an internal database which contains information on the structure of the current hyperobject. In particular, this database contains information recording all faces and hyperfaces of the hyperobject in terms of the vertex points and connecting edges that comprise them. When a single vertex point is moved arbitrarily in hyperspace, it will in all likelihood no longer be located exactly on any face planes or hyperface hyperplanes to which it actually belongs, and this has the potential for distorting the mathematical correctness of any geometrical calculations carried out in generating the displayed image. The upshot of all this is that the translation of a vertex is always treated as a small and temporary local perturbation of the overall geometry. HYPERGEO remembers the original (true) position of the vertex, and this position is always automatically restored as soon as any command is invoked that may entail a permanent transformation of the database. The intent of the vertex translation feature is to provide the ability to temporarily "tweak" a particular vertex and view how that perturbation is reflected in the current image. It does not allow the construction of a new, permanently modified geometry definition. 3.3.4.1 4-D Image Considerations To be able to select and translate a particular vertex of the current hyperobject, the point corresponding to that vertex must be visible in the current display. It should be recalled that in the case of images generated by performing an intersection of the hyperobject with a hyperplane, the points and edges seen in the final display do not correspond directly to points and edges in the hyperobject. Such a correspondence is only certain to exist with images generated as projections of the hyperobject onto a 3-D hyper-viewplane. For this reason, the translation state of HYPERGEO Version 2.1 Page 3-23 the user interface can only be entered when the display is of a 4-D to 3-D projection image. However, it should be noted that this includes the dual imaging mode in which both an intersection and a projection are shown simultaneously on the screen side by side. With a dual image, the projection portion can be used to select a vertex with the mouse, and the results of translating the vertex appear as temporary perturbations of both images. In fact, this probably makes the dual imaging mode the most interesting and useful one for performing vertex translations. 3.3.4.2 The Two User Interface States: Rotation and Translation As mentioned in the preceding section, the normal HYPERGEO user interface provides commands to perform rotations of the current geometry. An interactive command exists (keystroke 'U') to toggle the user interface between this default state and the alternative state in which an individual vertex point may be translated in hyperspace. Which state the user interface is in is indicated by the commands available in the menu area and by the behavior of the mouse cursor. In the default rotation state of the user interface, the mouse cursor is an arrow icon, and the cursor is always restricted to the menu area (if displayed). When the user interface is switched to the translation state, the cursor becomes a cross-hair icon, and it is repositioned to the main graphics area. While the cursor is a cross-hair, the mouse can be used to select a vertex point which is to be translated. Once a vertex has been selected with the mouse, it is marked in the display with a cross-hair symbol, and the mouse cursor reverts to the normal arrow icon and returns to the menu area. The special translation state commands of the user interface can then be used to perform temporary movements of the selected vertex. The contents of the bottom portion of the menu area depend upon the user interface state. In the normal rotation state, this part of the menu contains the mouse control pads for performing 4-D planar rotations: ____________________ | | | | | | | | | YZ | XZ | XY | | | | | |______|______|______| | | | | | | | | | XW | YW | ZW | | | | | |______|______|______| | | | 4-D PLANE | |____________________| HYPERGEO Version 2.1 Page 3-24 When the translation state of the user interface is in effect, 4-D rotations are unavailable, and the bottom portion of the menu area is replaced with a control pad containing the vertex translation functions: ____________________ | | | | | | | | | Sel | Y | Nxt | | | | | |______|______|______| | | | | | | | | | Z | W | X | | | | | |______|______|______| | | | MOVE VERTEX | |____________________| The pads labeled X, Y, Z, and W perform a translation of the vertex along the corresponding coordinate (the right mouse button increments the coordinate value, and the left mouse button decrements it). Selecting the pad labeled "Sel" deselects the current vertex and sets the interface to select a new one; the mouse cursor changes to a cross-hair icon and moves to the main display area. Selecting "Nxt" deselects the current vertex and automatically selects the next sequential one according to an (essentially arbitrary) ordering scheme used by the internal database. Whenever a vertex that has had temporary translations performed on it is deselected (either via the "Sel" or "Nxt" functions described above, or by toggling out of the translation state of the user interface entirely via the 'U' command), the permanent 4-D coordinates of the vertex are restored. 3.3.4.3 Vertex Translations without a Mouse Keyboard equivalents enable the performance of vertex translations even if no mouse is present on the system. Only the function of selecting a particular vertex in the display cannot be performed directly (although the same effect can be achieved in a somewhat roundabout fashion). HYPERGEO Version 2.1 Page 3-25 Without a mouse, when the program enters the translation state of the user interface (via the 'U' command), there is no visible and movable cursor for selecting a vertex. However, the <Enter> key will select one vertex arbitrarily. Once a vertex is selected, <Enter> is equivalent to the "Nxt" function pad in the menu area; it allows cycling through all vertexes in an arbitrary order. The <Esc> key is equivalent to the "Sel" function; it deselects the current vertex. The arrow keys can be used to perform the actual vertex translations. Right-arrow translates the selected vertex along the X-coordinate, up-arrow along the Y-coordinate, left-arrow along the Z-coordinate, and down-arrow along the W-coordinate. Using the arrow keys without any modifer key causes translations in the positive direction along the particular axis. Combining the arrow keys with the <Shift> key performs the translation in the negative direction. (Because <Shift> does not work with the arrow keys correctly on some keyboards, <Ctrl> can also be used wth the arrow keys to perform translations in the negative direction.) 3.4 View Control The final step in the processing chain that leads from the specification of the geometry of a four-dimensional hyperobject to the depiction of an image of that object on the computer screen is the generation of the two-dimensional graphical output. As described above in a previous section of this document, the program uses a number of variations of a basic 3-D to 2-D linear projection to perform this step. The user has access to several interactive commands with which to adjust a number of parameters that affect the appearance of the final display. Paramount among these is the output display mode. 3.4.1 The Five Display Modes As outlined earlier, there are five possible projection modes in which the final graphical output can be presented: perspective, stereoscopic anaglyph, hidden-line, surface relief, and solid. The selection of a particular mode is one of the parameters which the user can modify; any one of the five modes can be invoked at any time (with some exceptions - see below). A data field in the menu area shows which mode is in effect. Whenever the display mode is changed, the program immediately redraws the representation of the current geometry using the method of the new mode. The basic orientation of the hyperobject remains fixed across any such switch in display mode. HYPERGEO Version 2.1 Page 3-26 The exceptions referred to in the preceding paragraph are as follows: first, the stereoscopic anaglyph and surface relief modes cannot be invoked on monochrome systems (since the anaglyph requires red and blue); and second, the hidden-line, surface relief, and sold display modes can only be selected if the 4-D to 3-D reduction step is using 4-D intersection, not 4-D projection (since 4-D projections do not produce a 3-D geometry which is a solid 3-D polyhedron amenable to hidden-line analysis). 3.4.2 Key Display Parameters Two numeric parameters are key in the calculations which generate the graphics seen on the screen. First, the eye-to-screen distance is used to calculate all forms of 3-D to 2-D projection. The eye-to-screen distance is shown as a data item in the menu area. It can be adjusted up or down interactively by the user. Decreasing the eye-to-screen distance has the effect of enhancing the perceived degree of perspective in the picture, and increasing the distance reduces the perspective. The eye-to-screen distance can be adjusted in increments or decrements equal to 10% of its current value, or it can be set to an explicit numeric value. The program automatically limits the maximum and minimum values to which the eye-to-screen distance can be set in order to prevent numeric errors from occurring in calculations that are performed when generating the perspective displays and which make use of the eye-to-screen value. The second key display parameter is the scale factor which is the ratio of screen inches to geometry units. This value determines the apparent size of the object as finally drawn on the screen. The scale factor is displayed as a data field in the menu area. It can be adjusted up or down in increments of 10% of the current value, or it can be assigned an explicit numeric value. When a new geometry file is read in, and if there is no overriding information in the configuration file, the program automatically sets the scale factor so the image of the object fits comfortably within the primary graphics area allowing plenty of room for rotations. Both the eye-to-screen distance and the display scale factor can be specified in the configuration file. HYPERGEO Version 2.1 Page 3-27 3.4.3 Hyperface Highlighting The concept of a four-dimensional solid hyperobject's possessing three-dimensional hyperfaces (a higher-order geometrical analogy to the intuitively comprehensible fact that a three-dimensional polyhedron possesses two-dimensional faces) is naturally very difficult for us to grasp. To aid in visualizing this, HYPERGEO has a feature which enables the user to highlight that portion of the displayed 3-D image (either projection or intersection) that belongs to a single hyperface. A special highlight color (distinct from the primary graphics foreground color) is used to display all edges that are part of the hyperface that is currently highlighted. This makes the shape and topological structure of the hyperface readily visible as the hyperobject in its entirety is manipulated in hyperspace. It should be noted that a different interpretation must be placed on the visible pattern of highlighted edges depending on whether the 3-D image is a projection or an intersection. For a 4-D to 3-D projection, the edges seen in the final displayed image all correspond exactly to actual edges that exist in the hyperobject. Since a hyperface must be a simple, convex polyhedron, the pattern of hightlighted edges will also always form a complete polyhedron (although its shape will appear variously twisted and distorted due to the effects of the 4-D to 3-D projection). For 4-D to 3-D intersections, the situation is different. The edge lines that appear in the final display for an intersection correspond not to the original edges of the hyperobject but rather to the outlines of the polygons that are formed at the intersections of each of the hyperobject's hyperfaces with the intersecting hyperplane. Thus, a highlighted hyperface in a 4-D to 3-D intersection image will appear as a single highlighted face of the three-dimensional polyhedron that is the overall intersection. It is also possible that, as the hyperobject is rotated in 4-space and the intersecting hyperplane is moved, the highlighted hyperface may for a time fail completely to intersect the hyperplane. When this happens, no highlighted edges will appear in the 3-D intersection image (since the 3-D polyhedron has at that moment no face derived from an intersection of that particular hyperface). Indeed, it is generally most revealing to observe the changing (and occasionally vanishing) shape of the highlighted face that is the intersection of the hyperplane and the highlighted hyperface as the hyperobject is manipulated in 4-space. The <Tab> key is used to toggle hyperface highlighting on and off. When highlighting is on, the <Space> bar advances the highlighting to the next hyperface (according to an arbitrary internal ordering), and <Backspace> highlights the preceding hyperface (per the same arbitrary ordering). HYPERGEO Version 2.1 Page 3-28 Ordinarily, highlighting uses the color which is specially designated for that purpose in the overall HYPERGEO palette (see the discussion on HYPERGEO's use of color in Section 3.2.1). However, in certain circumstances, a different color or technique is used. On monochrome systems a thick line is used to indicate highlighting rather than a different color. In the display modes that use red-blue anaglyphs (stereoscopic and surface relief), a brighter intensity of the requisite red and blue shades is used for the highlighted edges. And when hyperface highlighting is used together with the W-coordinate spectrum banding feature (see below), the normal foreground color (default white) is the highlight color. 3.4.4 W-coordinate Spectrum Banding It is inherent in the nature of 4-D to 3-D image reductions (especially projections) that the W-coordinate information of the original hyperobject is lost. This follows inevitably from the fact that the projection takes place onto a hyperplane that has a constant W-coordinate value. The resulting image has full 3-D structure; the X and Y dimensions are readily apparent, even after projection onto the 2-D viewscreen, and the Z dimension can be perceived to a greater or lesser extent (most greatly in the stereoscopic projection modes). However, the W dimension gets lost in the reduction process and must be inferred indirectly from the changing appearance of the 3-D image as the hyperobject is manipulated. As an aid to perceiving the W-dimension information, HYPERGEO has a feature that causes all edges of the current image to be displayed in a color (or spectrum of colors) that is a function of the W-coordinate values of the corresponding edge of the original hyperobject. This feature uses a spectrum of ten colors in place of the single primary foreground color. The spectrum ranges from blue-violet for the minimum W-coordinate value, through blue, aqua, cyan, blue-green, green, olive-green, yellow, and orange, up to red for the maximum W-coordinate. The total range of possible W coordinates is divided into ten bands, and each band is assigned its corresponding spectrum color. All edge lines are displayed divided into separate segments according to how the W-coordinate values of the original 4-D edge range across the spectrum bands; each segment is displayed in the color that is correct for that segment's W-coordinate values. HYPERGEO Version 2.1 Page 3-29 For an image that is a 4-D to 3-D intersection, it should be noted that, by definition, all visible lines in the image have a single W-coordinate value (i.e., the current W-value of the intersecting hyperplane). Therefore, with W-coordinate spectrum banding in effect, an intersection image will be displayed entirely in whatever spectrum color corresponds to the W coordinate of the intersecting hyperplane. This will not change if the hyperobject is rotated, but it will change when the intersecting hyperplane is translated along the W-axis. In this latter case, the 3-D image will continue to be a single color, but that color will change through the spectrum shades as the hyperplane's W-coordinate ranges across the ten possible bands. The spectrum banding feature is toggled on and off with the 'B' key. 3.5 Journalling The journalling feature in HYPERGEO permits the recording into a disk file of all display information that appears on the screen during a sequence of manipulations of the current geometry. A recorded journal file can later be replayed, redisplaying the exact images that were visible during the original interactive session. This ability to record and replay a sequence of displayed images can be useful in several ways. By saving the display to a journal, the user can be sure of being able to recreate a particularly interesting image that might happen to occur and would not otherwise be easily reproducible. Also, the automated rotation commands (in particular, the commands that peform sequences of unpredictable - and unrepeatable - random rotations) can have their display sequences recorded in a journal file, thus ensuring that they can be redisplayed at will exactly as they originally appeared. Finally, for very large geometrical models, where the computational time spent in updating the database between images is significant, the display can be recorded, and when the journal is replayed, the computational delays are eliminated; the recorded images can be displayed as fast as the computer's graphics system will allow. This enables a relatively slow and long-running interactive sequence to be viewed at a speed that makes the behavior of the geometry far more intuitively comprehensible. A word of warning is in order here. Although the data recorded in a journal file is reduced to the minimum amount necessary to recreate the display, the volume of data required can still be considerable, especially for large and complex geometries. It is very possible that a journal file, if left recording over an extended period of time, will fill megabytes of disk space. (The W-coordinate spectrum banding feature described above is HYPERGEO Version 2.1 Page 3-30 especially costly in this regard since it multiplies the number of individual segment records, with their coordinate and color information, that must be written to the journal file.) Therefore, unless you have enormous ammounts of disk space at your disposable, it is wise to first create a brief trial journal of a particular geometry to get an approximate idea of how many bytes are needed for each separate recorded image. You also should be aware of how much free disk space your system has available (the DOS command "DIR" will tell you this). Because the data contained in a journal file reflects only the actual display seen on the screen, it is specific to the current system configuration. Journal files are, in general, not portable between systems. For example, a journal recorded in monochrome will definitely not look right if replayed on a color system. In addition, it is just possible that a journal file will not work correctly on the same system if the HYPERGEO configuration is changed significantly between the journal's recording and its replaying. 3.5.1 Recording a Journal File Recording of a journal file may be started any time that a geometry is active in the HYPERGEO display. The keystroke 'K' is used to start recording. A text dialog box appears which requests input of the name of the journal file to be created. When the dialog box first comes up, the type-in area contains a default file name which is the name of the current geometry definition with an extension of .JNL. The text editing features of the dialog box can be used to modify this file name if desired, including the prepending of a DOS path specification to place the file in a different directory. Once the journal file name has been confirmed, recording is in effect. A message indicating that a journal file is being recorded appears at the bottom of the main display area. Every interactive operation that changes the display on the screen writes the new image to the journal file. Within the journal each separate image is known as a "frame". Only the primary graphics are written to the journal; the menu area is not journalled, nor are any message or interactive dialog boxes that might appear as various HYPERGEO commands are executed. The keystroke 'K' is also used to terminate the recording of a journal file. It brings up a message dialog box that shows the name of the journal file, the date and time its recording was started, and the total number of frames it contains. HYPERGEO Version 2.1 Page 3-31 3.5.2 Replaying a Journal File A previously recorded journal file can be replayed whenever HYPERGEO is accepting interactive commands. For the replaying of a journal it does not matter what the current geometry is since the journal will override the current display while it is being replayed, and the current interactive display will be restored when the journal replay is finished. The keystroke 'J' invokes the journal replay command. A file selection dialog box appears listing all files that have extension .JNL in the current directory. The features of the dialog box can be used to select a journal file which is located in a different directory or which has a non-standard extension (i.e., other than .JNL). Once a journal file name has been selected and confirmed, the program enters journal replay mode. The display of the current geometry is erased, and the first frame of the journal file is displayed. The name of the geometry file which was recorded appears in the upper-left of the main display area along with a frame counter that shows the current journal frame number along with the total number of frames in the file. The journal replay command has options for displaying the contents of the journal that are modeled on the controls available on a video tape recorder (VCR). They permit viewing the journal's frames in the normal forward direction and also in the reverse direction. The journal may be moved forward or backward through the frames at high speed as in a VCR's fast forward and fast reverse (rewind) commands. In addition, the display of the journal may be stopped on any frame, and the journal may be stepped forward or backward a single frame at a time. A menu line appears at the bottom of the screen during journal replay that lists the keystroke commands that perform these VCR functions: HYPERGEO Version 2.1 Page 3-32 F - Fast forward Advances the journal forward through the frames at high speed until the last frame is reached, then stops. The main display is blank during fast forwarding; only the frame counter is updated. P - Play Displays the journal file's frames sequentially in forward order until the end of the journal is reached, then stops. A - Advance Steps the journal forward one frame and stops. S - Stop Stops the journal on the current frame; that frame's display will remain visible indefinitely until another replay command is executed. B - Back Steps the journal backward one frame and stops. R - Reverse Displays the journal file's frames sequentially in reverse order until the start of the journal is reached, then stops. W - Rewind Advances the journal backward through the frames at high speed until the first frame is reached, then stops. The main display is blank during rewinding; only the frame counter is updated. E - Eject Ends the journal replay. The display of the current geometry is restored, and normal HYPERGEO interactive commands are in effect. The Play and Reverse commands display the journal file frames as rapidly as the system's graphics capabilities will allow. Normally, this is considerably faster than the speed of the interactive manipulations that were recorded in the journal. HYPERGEO Version 2.1 Page 3-33 An alternative set of control keys is also enabled during the replay of a journal file that are designed to be used on keyboards which have a numeric keypad. (Note: NumLock must be off to use these keypad commands.) They are entirely equivalent to the command keys listed above: Fast forward - End Play - Right-arrow Advance - Down-arrow Stop - '5' on the keypad Back - Up-arrow Reverse - Left-arrow Rewind - Home Eject - Escape A diagram of the keypad gives a better sense of the logic behind this arrangement: ________________________________ | | | | | | | | | Rewind | Back | | | | | | |__________|__________|__________| | | | | | | | | | Reverse | Stop | Play | | | | | |__________|__________|__________| | | | | | | | | | Fast Fwd | Advance | | | | | | |__________|__________|__________| As an additional convenience, the <Space> bar also executes the Stop command. HYPERGEO Version 2.1 Page 3-34 3.6 PCX File Output The display currently visible on the screen may be saved in a PCX file at any time during interactive editing. PCX is a standard graphics file format that can be read by many commercial word-processing, desktop publishing, and presentation artwork programs. This provides a means for copying HYPERGEO images into documents and for producing printed output. The 'O' key executes the PCX File Output command. It brings up a text dialog box which requests the name of the file to be created. The dialog box's type-in area is pre-filled with a default name which is the name of the current geometry file with .PCX as its extension. The dialog box features may be used to change this name if desired, including prepending a DOS path specification to place the output file in a different directory. Once the file name is confirmed, the PCX file is created. A message window appears indicating the successful completion of the command. HYPERGEO Version 2.1 Page 4-1 Section 4. Inner Workings 4.1 Units The coordinates of the geometry definition are considered to be in so-called "geometry units" which are not assumed to have any particular relation to any real physical unit. Regardless of the size of the coordinate values, HYPERGEO will scale the display to fit within the visible screen area. Some of the parameters that can be specified by the user are expressed in geometry units and must be entered that way. Other parameters, in particular, those relating to the projection of the final 2-D image on the computer screen, are expressed in real-world units. The program uses inches for all real-world dimensions, and the display scale is expressed as a ratio of inches to geometry units (the inches here refer to distances on the face of the screen). In the detailed reference material contained in the appendixes to this document and in all program prompts and messages the units to use for a particular data value are always indicated. 4.2 Verification of the Geometry Definition A geometry definition file is created by the user as an ordinary text file (see Appendix A for a full discussion of this file's format). It is then processed into a more efficient binary (i.e., non-text) form which can in turn be read by the HYPERGEO program. This text-to-binary processing step is performed by the GEO2BIN utility program. In addition to converting the geometry definition to binary format, the GEO2BIN program also performs some error checking on the correctness of the geometrical model. It analyzes the definition of the described object which contains direct information on the number of vertexes and edges. The number of faces and (for 4-D) hyperfaces is calculated, and various aspects of the input file and its derived topology are checked for validity: * The number of dimensions must be either 3 or 4. * Every vertex must have at least as many edges connected to it as the number of dimensions. * All edge connections must be consistently specified between both connected vertexes, that is, if vertex #1 is in the connection list of vertex #2, vertex #2 must be in vertex #1's list, also. * The numbers of vertexes, edges, faces, and (for 4-D) hyperfaces must agree with the theoretical values as given by Euler's Formula or its hypergeometric version. HYPERGEO Version 2.1 Page 4-2 If any of these checks fails, an error message is displayed, and the binary geometry definition file is not created. 4.3 Overlay Swapping HYPERGEO was developed using Borland C++ Version 3.1, and it makes use of the Borland overlay manager (called VROOMM for Virtual Run-time Object-Oriented Memory Manager). Various portions of HYPERGEO's executable code are swapped in and out of memory dynamically as they are needed while the program is running. This technique is known as overlaying and is a standard approach for minimizing a program's run-time memory requirements. It reduces the amount of system memory required by the program at any one time. However, overlaying has the drawback that it may entail repeated disk reads of the program's executable file (whenever a new portion of the code must be loaded for execution). This can cause a noticeable slow-down in the program's performance. A way to avoid this problem is to use system memory outside the standard 640K DOS area to store those portions of the program that are temporarily swapped out. When they are needed again for execution, they can be retrieved from memory much faster than if they must be reread from disk. The Borland overlay manager allows using either expanded memory (EMS) or extended memory as the overlay storage area. A full discussion of what expanded and extended memory are and how they are implemented on a PC is beyond the scope of this document. Those topics are covered in some detail in the DOS reference manual. Suffice it to say for now that if your computer has any memory beyond the 640K main DOS area, it is probably set up to be available as one or the other of expanded or extended. If your system has extra memory but you're not sure how it is set up, you can simple try running HYPERGEO with each of the two overlay swapping options (-e for EMS expanded, -x for extended) and see which one works. If you choose the wrong one, HYPERGEO will issue a message saying it can't find the specified type of memory and will not start up. Note: If your system has a disk caching utility installed (for example, Microsoft's SMARTDRV, which is provided with MS-DOS and Windows 3.x), you should not enable HYPERGEO's overlay swapping. The disk caching performs a similar function, and HYPERGEO's overlay swapping would be redundant. HYPERGEO Version 2.1 Page 5-1 Section 5. Restrictions and Limitations 5.1 Geometry Definition File The database maintained by the HYPERGEO program is dynamically allocated and expands to fit the size of the geometry being handled so there are no precise and absolute capacity limits on how big and complicated an object may be. The ultimate barrier is the 640K limit of available memory under DOS. Although I have not done the actual experiment, my guess is that the two largest regular four-dimensional polytopes, with 120 and 600 hyperfaces respectively (and as many as 1200 edges and faces), would far overwhelm HYPERGEO and DOS with their memory allocation requirements. If some intrepid researcher would care to undertake the creation of geometry definition files for these two very large hyperobjects, I would be most interested in hearing about the results. For more reasonably sized geometries, the most important restriction is that all 4-D hyperobjects must be "simple" and externally convex. (Simple means the object must be solid without any holes passing through it.) For 3-D geometries, the requirement that the object be externally convex is removed, but it must still be a simple polyhedron, and all its faces must be simple polygons without internal edges. Note that this last condition implies more severe limitations than is at first evident since it means that no face can have isolated projections or depressions in it (as that would mean there would be a ring of edges internal to the face and disconnected from the rest of the topology). As mentioned above, GEO2BIN performs some checking on the topological correctness of the input geometry. However, to be truthful, this checking is not totally comprehensive; it can certainly fail to detect some errors in the input, and when it does find errors, the messages issued are not always as helpful as could be wished in pinpointing the problem. I am sure that it would be possible to get the program to misbehave quite seriously, including hanging or crashing, by feeding it pernicious input data. Therefore, until a future version of the program possesses improvements in this regard, the user must take some pains to insure that all new geometry definition files are created carefully and accurately. If the program crashes upon startup, consider that to be a very loud, broadband error message, and check the geometry file closely. (However, HYPERGEO and the utility GEO2BIN are not totally deficient in this regard, and they will issue error messages when certain problems are detected. A geometry definition file with only a handful of conventional errors will be handled adequately.) HYPERGEO Version 2.1 Page 5-2 5.2 Mathematical Precision Most calculations in HYPERGEO are performed using a special fixed-point format that uses 32-bit data fields. Within this field, 24 bits are used for the fractional portion of the value, and this gives about seven decimal digits of accuracy. This is sufficient to produce graphics that are accurate and correct to a degree well beyond the limiting resolution of PC display devices. It is possible that small discrepancies may appear in some of the parameter values displayed by HYPERGEO as a result of restricting the precision of numerical calculations to 24 bits. For example, you may notice that if you perform a single rotation of the geometry away from its starting orientation (where all axis angles are exactly zero) and then immediately perform a single identical rotation in the reverse direction, the axis angles may not all be restored to precisely zero. This particular phenomenon is an especially sensitive indicator of very slight imprecisions in the fixed-point calculations; it arises because the axis angles are calculated as inverse cosines of orientation matrix coefficients, and the cosine function has the property that its value is very close to 1.0 for a fairly wide range of small angle values. (For example, the inverse cosine of 0.999999 is approximately 1/10 degree.) Despite such apparent inaccuracies, the calculations are accurate to a degree of precision more than sufficient to produce exactly correct graphics. All numbers that are accepted as user input to the program (such as values appearing in configuration file parameter specifications) are handled as double precision floating-point quantities. The double precision word length is 64 bits which provides about fifteen decimal digits of accuracy. Any floating-point values given to the program (such as coordinates in the geometry definition file or scale factor in the configuration file) may contain up to fifteen digits although the usefulness of such extreme precision is questionable. HYPERGEO Version 2.1 Page 6-1 Section 6. Supplied Geometry Definition Files The HYPERGEO program comes with several "ready-to-run" geometry definition files, including both 4-D and 3-D objects. These files include four of the six regular convex four-dimensional polytopes and all five of the three-dimensional regular convex polyhedra (the so-called "Platonic solids"). (As indicated above, the remaining two regular convex 4-D polytopes are almost certainly too big and complex to fit in HYPERGEO under DOS; they also are too far beyond my powers of hypergeometric comprehension to capture in a HYPERGEO input file.) All of the supplied geometry definition files are in binary format and have the file extension .BIN. The root portion of the file name corresponds to the name of the geometric object contained in the file (with some abbreviations as required by DOS file name length limits). The regular four-dimensional polytopes are in the files: File Name Vertexes Edges Faces Hyperfaces ------------ -------- ----- ----- ---------- 5CELL.BIN 5 10 10 5 HYPRCUBE.BIN 16 32 24 8 16CELL.BIN 8 24 32 16 24CELL.BIN 24 96 96 24 Note that except for the hypercube the 4-D polytope names are of the form 'N-cell' where N is the number of hyperfaces. (The hypercube, technically and to be consistent, could be called the 8-cell - just as the cube could be called a 'hexahedron' and the square a 'tetragon'.) For the record, the two 4-D regular polytopes that aren't included here are the 120-cell and the 600-cell. HYPERGEO Version 2.1 Page 6-2 For three dimensions, all of the regular polyhedra names except "cube" are of the form '<prefix>hedron' where <prefix> is the Greek numeral indicating the number of faces: tetrahedron, octahedron, dodecahedron, and icosahedron. The HYPERGEO geometry definition file names are formed from just the prefix without the base 'hedron': File Name Vertexes Edges Faces ------------ -------- ----- ----- TETRA.BIN 4 6 4 CUBE.BIN 8 12 6 OCTA.BIN 6 12 8 DODECA.BIN 20 30 12 ICOSA.BIN 12 30 20 In addition to the regular polytopes and polyhedra, the following geometry definition files are also included: File Name Dimensions Geometrical Object ------------ ---------- --------------------------------- BLOCKS.BIN 3 A three-dimensional cross formed by stacking cubes along each of the three coordinate dimensions. STELLA.BIN 3 The "stella octangula", an interesting polyhedron formed by merging two 3-D tetrahedra. HG.BIN 3 A shape that forms the HYPERGEO logo by presenting the outline of the letter 'H' when viewed from two of the three orthogonal directions and the outline of the letter 'G' when viewed from the third. HYPERGEO Version 2.1 Page 6-3 HYTETRA.BIN 4 These four objects are HYOCTA.BIN 4 "hyperprisms" derived from the HYDODECA.BIN 4 3-D regular polyhedra contained HYICOSA.BIN 4 in their names. A hyperprism is formed by positioning two 3-D objects in hyperspace at a given separation along the W-axis and connecting each of the pairs of corresponding points between the two objects. The resulting hyperobject has two hyperfaces that are the same shape as the base 3-D object plus an additional number of hyperfaces equal to the number of faces in the base 3-D object. These additional hyperfaces are 3-D prisms whose end faces are the same polygonal shape as the faces of the base 3-D object. Thus, for example, the HYTETRA.GEO hyperobject is based on the 3-D tetrahedron; it has two hyperfaces that are tetrahedra and four hyperfaces that are prisms with triangular end faces. (Note: Technically the hypercube is itself a hyperprism - one based on the cube - but since it is also one of the regular 4-D polytopes, it is not included in this series.) ADDITIONAL GEOMETRY DEFINITION FILES SUPPLIED WITH REGISTERED COPIES OF HYPERGEO In addition to those files listed above, a number of extra geometry files are included with the utility program material that is supplied with a registered copy of HYPERGEO. First, the text-format version (extension .GEO) of each of the .BIN files listed above is supplied. This text version is not needed to use the geometry file in HYPERGEO, but it is of value as documentation showing how the objects and hyperobjects are constructed. The text files may also be of use as models to follow if the user creates any additional geometry definitions from scratch. HYPERGEO Version 2.1 Page 6-4 Second, the following geometry definition files are supplied in both binary and text format (all are 3-D): File Name Geometrical Object ------------ ------------------------------------------------- CUBOCTA The cuboctahedron. This is a "quasi-regular" convex polyhedron which is formed as the intersection of a cube and an octahedron. The faces are not all identical as in the Platonic solids, but they are all regular polyhedra (squares and equilateral triangles), and they are cyclically regular about each vertex. STELCUBO The stellated form of the cuboctahedron. It is the union of a cube and an octahedron. RHDODECA The rhombic dodecahedron. This is the dual of the cuboctahedron. All of its faces are identical rhombuses. ICODODEC The icosidodecahedron, the second quasi-regular convex polyhedron. It is formed as the intersection of an icosahedron and a dodecahedron. Its faces are regular triangles and pentagons, and they are cyclically regular about the object's vertexes. STELICOS The stellated icosidodecahedron. It is the union of an icosahedron and a dodecahedron. TRIACONT The triacontahedron, which is the dual of the icosidodecahedron. It is also a rhombohedron. HYPERGEO Version 2.1 Page A-1 Appendix A. Geometry Definition File This appendix describes the text format of the geometry definition file. This file is the primary input to the HYPERGEO program. It specifies the topology and dimensions of the geometrical object to be displayed. In particular, the file contains the coordinate values for all vertex points of the object plus connectivity information that indicates which vertexes are joined by edges. Section A.1 of this appendix is a complete reference on the format of this file. A geometry definition file cannot be used directly by HYPERGEO in its original text form. It must first be converted into a binary format with the GEO2BIN utility program. The binary version of the geometry file can be read much more quickly by HYPERGEO than the text version since it contains the additional derived topological information required to construct the internal HYPERGEO database. Section A.2 of this appendix describes the use of the GEO2BIN utility. A.1 Text Format of the Geometry Definition File The geometry definition file is an ordinary text file that can be edited using any available text editor. It is read by the GEO2BIN program a line at a time; each line comprises one complete data record, and there is no way to continue data across multiple lines. The maximum length of a line is 255 characters. The geometry definition file can contain notes and comments that are not part of the data read by the program, but serve as human-readable annotation. This is indicated by preceding any such comments with the character '!'. GEO2BIN ignores any '!' character plus whatever follows it on the same line. If the first character on a line is a '!', the entire line is skipped by the program. If the '!' character appears following some data on a line, that data is read by the program, but everything after the '!' is ignored. The geometry definition file can contain any number of blank lines, which are ignored. A line which contains only comment text is considered by the program to be a blank line. Every geometry definition file must begin with two particular data lines. The first contains a single integer value which is the number of dimensions of the geometric object; this number must be either 3 or 4. The second line contains a single integer that is the number of vertex points in the geometry. Following these two lines, there must be one line with point data for each vertex point. HYPERGEO Version 2.1 Page A-2 Each point data line contains the coordinate and connectivity information for a single vertex point. Its format is: n: x.x y.y z.z [w.w] p1 p2 p3 [... pN] where 'n' is the sequence number of the point data line; 'x.x', 'y.y', 'z.z', and 'w.w' are the X, Y, Z, and W coordinates of the point; and 'p1' through 'pN' are the sequence numbers of all other points connected to this point by edges. Each point's sequence number serves to identify the point within the file. It is the number used in the connection lists, and it is also the number used to refer to the point in any error messages that may be issued. All point sequence numbers are integers, and they must be in ascending order within the file; the first point data line must have sequence number 1, and the last point data line must have a sequence number equal to the total number of vertex points in the object. The coordinate values are floating-point numbers and may contain an optional decimal point and fractional digits. They may also contain a decimal exponent in standard scientific notation format. The coordinate values may be in any units desired; the program always scales the final display to fit the screen. The W coordinate must be specified for a four-dimensional geometry and must not be specified for a three-dimensional one. The list of connected points contains the sequence numbers of all points connected to the point described by the given data line. Connection information must be symmetrical, that is, if point A is in point B's connection list, point B must also be in point A's connection list. It is an error for a point to be in its own connection list. Unlike the coordinate values, the program has no way of knowing how many connections should exist for each point and hence cannot directly report an error if there are too few or too many. Therefore, it is incumbent upon the user to construct these lists with some care. Except for the ':' that must follow the sequence number, no punctuation is used to separate the data values, just one or more spaces. Following all point data, the geometry definition file may optionally contain configuration information. If it appears, it must consist of statements that are identical in format to the primary configuration file (see Appendix B). Configuration parameters specified in a geometry definition file override corresponding parameters set in the primary configuration file, but they remain in effect only while that geometry definition file is active in the HYPERGEO program. HYPERGEO Version 2.1 Page A-3 A.2 The GEO2BIN Utility Program Every text Geometry Definition File (default extension .GEO) must be pre-processed to produce the corresponding binary version of the geometry definition (extension .BIN) in order to be used as input to the HYPERGEO program. The GEO2BIN utility program is provided to perform this text-to-binary conversion step. It must be executed (once only) for each new text geometry file before the file can be used in HYPERGEO, and it must be executed again (once) whenever any change is made to the text version of the geometry file. Note that this includes changes made to any configuration data that may be specified in the geometry definition file; the GEO2BIN program copies all such configuration data into the binary (.BIN) version of the file so it can be read by the main HYPERGEO program. GEO2BIN is a normal DOS program; it is executed by typing its name on the DOS command line followed by the name of the text geometry definition file to be processed: GEO2BIN <name-of-text-geometry-file> The geometry file name may contain a DOS path specification; otherwise, it is assumed to exist in the current directory. If the name as entered has no extension, .GEO is assumed. The output file created by GEO2BIN has the same name (and goes in the same directory) as the input text file, but with the extension .BIN. The GEO2BIN program displays status information indicating its progress as it is processing the geometry data. If GEO2BIN finds errors in the geometry file, it displays appropriate messages on the screen and does not create the binary output file. If desired, these error messages can be written to a file rather than displayed on the screen by using the standard DOS command-line redirection facility (the '>' character). Examples: GEO2BIN HYPRCUBE This reads the text geometry file HYPRCUBE.GEO in the current directory and creates the binary version HYPRCUBE.BIN in the current directory. GEO2BIN D:\DATA\CUBE.DAT >GEO2BIN.ERR This reads the text geometry file CUBE.DAT in directory D:\DATA and creates the binary version CUBE.BIN in directory D:\DATA. Any error messages will be written to the file GEO2BIN.ERR in the current directory rather than being displayed on the screen. HYPERGEO Version 2.1 Page B-1 Appendix B. Configuration File This appendix describes the format of the HYPERGEO configuration file. This file allows the user to set a number of parameters that control various aspects of the program's performance. The file is read once upon start-up of the program, and is reread each time a new geometry definition file is entered. In addition to the configuration file, similar configuration information may also be contained in any geometry definition file. The format is the same in both places, but the parameter specifications in the geometry definition file override those in the primary configuration file; however, they are in effect only while that geometry file is active in the program (see Appendix A). Note: There is one configuration parameter that is restricted to being used only in a geometry definition file and that may not be used in the primary configuration file. See the entry on the ORIGIN parameter in the list below. A command line argument exists that allows the user to specify the name of the particular primary configuration file to use. This permits multiple configuration files to exist and to be selected as desired for different executions of HYPERGEO. If no configuration file name is specified via a command line argument, the program tries to open one called HYPERGEO.CFG in the current directory. It is not an error if this default configuration file cannot be opened, but it is a fatal error if a configuration file whose name is specified by the command line argument cannot be opened. The configuration file is an ordinary text file that can be edited using any available text editor. It is read by the HYPERGEO program a line at a time; each line comprises one complete data record, and there is no way to continue data across multiple lines. The maximum length of a line is 127 characters. The configuration file can contain notes and comments that are not part of the data read by the program, but serve as human-readable annotation. This is indicated by preceding any such comments with the character '!'. HYPERGEO ignores any '!' character plus whatever follows it on the same line. If the first character on a line is a '!', the entire line is skipped by the program. If the '!' character appears following some data on a line, that data is read by the program, but everything after the '!' is ignored. HYPERGEO Version 2.1 Page B-2 The configuration file can contain any number of blank lines, which are ignored. A line which contains only comment text is considered by the program to be a blank line. Each data line in the configuration file contains a single parameter definition. The format of each line is simply the key word identifying the parameter followed by whatever data values the particular parameter requires. No punctuation should be used; all parameters and values should be separated by one or more spaces. The case of letters used in the configuration file does not matter, so lower or upper case (or both) can be used as desired. However, the program is absolutely unforgiving about spelling; all parameter key words must be spelled exactly as they are shown in this document - no abbreviations, no hyphens instead of underscores, no near misses. Most parameters take a single data value, either a number or another key word. Some parameters take a sequence of several numeric values, and some take no value at all. Numeric values that are inherently integral must be entered as pure integers. Where floating-point values are appropriate, the data item may include a decimal point and fractional digits as well as an exponent in scientific notation. Errors encountered by the program while trying to read a configuration file (or configuration data in a geometry definition file) are listed on the screen, and the program pauses to permit the user to read them. The primary configuration file is read by the main HYPERGEO program, while the secondary configuration specifications contained in a geometry definition file are read by the GEO2BIN program; but in both cases the method of error reporting is the same. All configuration file errors are fatal and terminate the program (HYPERGEO or GEO2BIN) after the error message list is dismissed. The following pages contain a list of all parameters that may be specified in the HYPERGEO configuration file or a geometry definition file. HYPERGEO Version 2.1 Page B-3 Parameter: COLOR COLOR_BACKGROUND COLOR_HELP_BACKGROUND COLOR_HELP_TEXT COLOR_HILITE COLOR_MENU COLOR_MESSAGE_BACKGROUND COLOR_VECTOR Value: A color name chosen from the sixteen standard EGA colors. The possible colors are: BLACK DARKGRAY BLUE LIGHTBLUE GREEN LIGHTGREEN CYAN LIGHTCYAN RED LIGHTRED MAGENTA LIGHTMAGENTA BROWN YELLOW LIGHTGRAY WHITE Use: These eight parameters set the colors used for each of the component parts of the HYPERGEO display: COLOR - Primary geometry graphics; also the transient text values in the menu area and text in dialog boxes COLOR_BACKGROUND - Screen background COLOR_HELP_BACKGROUND - Background color of the on-line help screen COLOR_HELP_TEXT - Text color of the on-line help screen COLOR_HILITE - Color used to display edges of a highlighted hyperface COLOR_MENU - Outlines and fixed labels in menu area plus the mouse control pad; also the geometry file name COLOR_MESSAGE_BACKGROUND - Background color for all message and prompting dialog boxes COLOR_VECTOR - Unit axis vectors The screen background color should be a dark color (BLACK or DARKGRAY) if the stereoscopic anaglyph mode of display is to be used. HYPERGEO Version 2.1 Page B-4 Default: COLOR WHITE COLOR_BACKGROUND DARKGRAY COLOR_HELP_BACKGROUND LIGHTGRAY COLOR_HELP_TEXT BLACK COLOR_HILITE LIGHTMAGENTA COLOR_MENU YELLOW COLOR_MESSAGE_BACKGROUND BROWN COLOR_VECTOR CYAN Parameter: COLOR_SOLID Value: A red/green/blue color specification in the form of three integers each in the range 0 to 63. The integers should be entered without punctuation, separated by spaces. Use: On VGA color systems, defines the colors to be used for surface rendering in solid display mode. For solid surface rendering the program uses twelve shades of graduated intensity based on the same color. The shades range from light to dark and are used to represent the varying degrees of illumination received by the different faces of the object from a theoretical light source. (HYPERGEO can optionally use color dithering techniques to increase the effective number of solid fill shades from twelve to twenty-three; see the SOLID_DITHER parameter below.) The color specified by the COLOR_SOLID parameter defines the lightest of the twelve shades; the program creates the remaining eleven to be increasingly darker variations. Therefore, the color specified should be fairly light, that is, have relatively high values for the red, green, and blue color components. Note: This parameter is only meaningful on systems with VGA color graphics. Default: 55 63 48 This defines a light, slightly grayish green. It produces a spectrum of twelve shades that range through medium gray-greens down to a dark, evergreen green. HYPERGEO Version 2.1 Page B-5 Parameter: DELTA_W Value: numeric value (floating-point) Use: Specifies the incremental step amount used to modify the W intersection value during execution of the HYPERGEO command for translation of the intersection hyperplane (Page Up/Page Down - see Appendix D on Interactive Commands below). This value is in the units of the geometry definition. Because this parameter is in units that pertain to a particular geometry definition, it is most useful in a geometry definition file; however, it may be specified in the primary configuration file. Default: 0.05 times the span of the geometrical object. The "span" is the greatest extent of the object in any one of its three or four dimensions. Parameter: DISPLAY_INFO Value: One of: OFF ON Use: Specifies whether or not the HYPERGEO information window should be displayed automatically at program start up and whenever a new geometry definition file is entered. Default: ON HYPERGEO Version 2.1 Page B-6 Parameter: DISPLAY_MENU DISPLAY_NAME DISPLAY_VECTOR Value: One of: OFF ON Use: Specifies whether a particular component of the HYPERGEO display will be drawn. DISPLAY_MENU controls the display of the menu area along the right edge of the screen. If DISPLAY_MENU is OFF, the entire screen is used for the primary geometry graphics (in this state, the mouse functions are not available). DISPLAY_NAME controls the display of the name of the geometry definition file; if ON, the name will be displayed at the upper, left corner of the screen. DISPLAY_VECTOR controls the display of the coordinate axis vectors. The axis vectors are displayed as dotted lines in the primary graphics; each vector is labeled with the name of its axis, X, Y, Z, or W. Note: Regardless of the DISPLAY_VECTOR setting, the axis vectors are not displayed in hidden-line, surface relief, or solid display mode. Default: DISPLAY_MENU ON DISPLAY_NAME ON DISPLAY_VECTOR OFF Parameter: DISPLAY_MODE Value: One of: ANAGLYPH HIDDEN_LINE PERSPECTIVE RELIEF SOLID Use: Specifies the display mode used to draw the primary geometry graphics. Note: The HIDDEN_LINE, RELIEF, and SOLID modes cannot be used with 4-D to 3-D projections (only intersections). If they are specified together with the PROJECTION or DUAL parameter (see below), the program will use PERSPECTIVE mode. Default: PERSPECTIVE HYPERGEO Version 2.1 Page B-7 Parameter: DUAL Value: None Use: Causes HYPERGEO to create dual images using both intersection and projection modes of 4-D to 3-D reduction; both images are displayed on the screen simultaneously side by side (the intersection is on the left and the projection on the right). The DUAL parameter is meaningless for three-dimensional geometries. DUAL, INTERSECTION, and PROJECTION are mutually exclusive alternatives; if more than one appears in configuration file data, the final one encountered will take precedence. Default: By default HYPERGEO uses intersection mode for 4-D to 3-D reductions. Parameter: EPSILON Value: numeric value (floating-point) Use: Specifies a small value to be used as an effective zero in all floating-point comparisons required by the program's calculations. This value is unitless; all calculations are performed internally on normalized coordinates. Default: 1.0E-6 Note: The performance of some of the algorithms used in hidden-line and solid surface displays is very sensitive to this value. The default was set after much experimentation and appears to provide reliable behavior almost all of the time. Users fiddle with this value at their own risk. HYPERGEO Version 2.1 Page B-8 Parameter: EYE_DISTANCE Value: numeric value (floating-point) Use: Specifies the viewing distance, that is, the distance between the eye and the display screen. This value is in inches. It determines the degree of perceived perspective in the various display modes. For greatest realism, it should be set to accurately reflect the true eye-to-screen distance for the user's normal viewing position. Decreasing the EYE_DISTANCE enhances the perspective effect, which can be useful. HYPERGEO automatically limits the maximum and minimum values to which the eye-to-screen distance can be set to prevent possible numeric errors in the program's computations. Default: 30.0 Parameter: EYE_SEPARATION Value: numeric value (floating-point) Use: Specifies the distance in inches between the viewer's eyes. This value is used in the perspective calculations required for the separate left and right eye images of the stereoscopic anaglyph and surface relief display modes. Default: 2.5 Note: This default value should be entirely acceptable for virtually all users. You should only try adjusting it if you are having difficulty fusing the two stereoscopic images into a single 3D view with proper depth perception. HYPERGEO Version 2.1 Page B-9 Parameter: INTERSECTION Value: None Use: Causes HYPERGEO to use the intersection mode of 4-D to 3-D reduction (rather than projection mode). The INTERSECTION parameter is meaningless for three-dimensional geometries. DUAL, INTERSECTION, and PROJECTION are mutually exclusive alternatives; if more than one appears in configuration file data, the final one encountered will take precedence. Default: By default HYPERGEO uses intersection mode for 4-D to 3-D reductions. Parameter: ORIGIN Value: Coordinate values as a sequence of three (for 3-D) or four (for 4-D) floating-point numbers. Values should be entered without delimiting punctuation, using only spaces as separation. Use: Permits translating the geometry along any (or all) of the coordinate axes. The coordinate values specified define the new origin which is to be used by HYPERGEO as a centerpoint for all rotations. The ORIGIN parameter may only be used in configuration data appearing within a geometry definition file; it may not be used in the primary configuration file. This is because the validity of the origin coordinates - in terms of their number and the units they are in - depends on the associated geometry definition. Also, the translation is applied dynamically to the geometry point definitions as they are read in. The ORIGIN coordinate values are in the same units as the original vertex point coordinates in the geometry definition file. Default: The origin is assumed to be (0,0,0) for 3-D and (0,0,0,0) for 4-D geometries. HYPERGEO Version 2.1 Page B-10 Parameter: PROJECTION Value: None Use: Causes HYPERGEO to use the projection mode of 4-D to 3-D reduction (rather than intersection mode). The PROJECTION parameter is meaningless for three-dimensional geometries. DUAL, PROJECTION, and INTERSECTION are mutually exclusive alternatives; if more than one appears in configuration file data, the final one encountered will take precedence. Default: By default HYPERGEO uses intersection mode for 4-D to 3-D reductions. Parameter: SCALE Value: numeric value (floating-point) Use: Specifies the scale factor at which the display is drawn. The units of this value are screen inches per geometry unit. Because this parameter is in units that pertain to a particular geometry definition, it is most useful in a geometry definition file; however, it may be specified in the primary configuration file. Default: Autoscale: the display is scaled to fit comfortably within the primary graphics area allowing ample room for interactive rotations. HYPERGEO Version 2.1 Page B-11 Parameter: SCREEN_HEIGHT SCREEN_WIDTH Value: numeric value (floating-point) Use: These two parameters specify the vertical and horizontal size in inches of the actual area on the display screen in which graphics are drawn. They can best be determined for a particular monitor by simply measuring the height and width of the displayed area on the face of the screen. The precise accuracy of these parameters isn't critical, but they are useful to insure that the program's display scale is consistent between the horizontal and vertical directions (so that squares look square and not rectangular, for example). Default: SCREEN_HEIGHT 6.0 SCREEN_WIDTH 8.0 Note: These default values are about right for the typical 12-inch monitor. Parameter: SOLID_DITHER Value: One of: ON OFF Use: Specifies whether or not the program will use color dithering techniques to extend the number of shades available for solid surface rendering. (Dithering is the pixel-by-pixel blending of different colors to produce what appears to the eye as an intermediate solid shade.) HYPERGEO generates a palette of twelve distinct shades that are used to fill the faces of an object viewed with the solid display mode. With dithering, this number is increased to twenty-three. The only disadvantage to using dithering is that it slows down the speed at which the graphics are drawn. Disabling this feature should only be necessary if display speed is unusually critical; otherwise, use dithering. Note: This parameter is meaningful only for systems with color graphics. Default: ON HYPERGEO Version 2.1 Page B-12 Parameter: SOLID_FILL Value: One of: COLOR PATTERN Use: Specifies whether to use color or pattern for surface rendering in solid display mode. When color is used for solid surface rendering, HYPERGEO displays the faces of the object with varying levels of brightness to indicate how they are illuminated by a theoretical light source. If pattern is used instead, the same effect is achieved by varying the density of fill (displayed with a single color) used to render the object's solid faces. If color is used, the twelve shades derived from the COLOR_SOLID color (see above) are used for VGA graphics, and a set of twelve pre-programmed colors are used for EGA graphics. Dithering may be used to increase the number of apparent shades to twenty-three (see the SOLID_DITHER parameter above). If pattern is used, it is displayed using the color specified for the primary geometry graphics. Note: This parameter is only meaningful on systems with color graphics capability (EGA or VGA); on monochrome systems, PATTERN is always used. Default: COLOR (for EGA and VGA color systems) PATTERN (for monochrome systems) Parameter: TURN Value: numeric value (floating-point) Use: Specifies the incremental step angle used by all interactive rotation commands. This value is specified in degrees. Default: 5.0 HYPERGEO Version 2.1 Page B-13 Parameter: VECTOR_LENGTH Value: numeric value (floating-point) Use: Specifies the length in screen inches of the unit axis vectors which may optionally be displayed as part of the primary geometry graphics (see DISPLAY_VECTOR above). The VECTOR_LENGTH is the length an axis vector appears on the screen when it is situated parallel to the XY-plane. As the object is rotated, the various axis vectors will appear to rotate with it and become visibly shorter as they assume different oblique orientations. Default: 1.0 Parameter: W_INTER Value: One of: CENTERED numeric value (floating-point) Use: Specifies the W-coordinate value at which a four-dimensional hyperobject will be "sliced" when using 4-D to 3-D intersection mode (that is, the value 'k' in the equation, w = k, of the intersecting hyperplane). If a numeric value is specified, it should be in the same units as the geometry definition. When CENTERED is specified, the program automatically calculates the W value to use as the average of the minimum and maximum of all W coordinate values in the geometry definition. This slices the hyperobject more or less down the middle. Because this parameter is in units that pertain to a particular geometry definition, it is most useful in a geometry definition file; however, it may be specified in the primary configuration file. Default: CENTERED HYPERGEO Version 2.1 Page B-14 Parameter: W_PROJECT Value: numeric value (floating-point) Use: Specifies the W-coordinate value at which the theoretical viewpoint is positioned in hyperspace for generating 4-D to 3-D projections. The value is in the same units as the geometry definition. Because this parameter is in units that pertain to a particular geometry definition, it is most useful in a geometry definition file; however, it may be specified in the primary configuration file. Default: 2 times the "radius" of the hyperobject HYPERGEO Version 2.1 Page B-15 The following is an example of a HYPERGEO configuration file: ! Example HYPERGEO Configuration File ! PROJECTION ! change from default Intersection Color_Background Black Color_Vector LightCyan ! lighter than default CYAN COLOR_SOLID 63 53 43 ! light reddish tan - gives ! darker shades that range ! through cocoa on down to ! dark red-brown display_vector on vector_length 2.0 ! up from default 1 inch TURN 45 ! for fast rotations DISPLAY_MODE anaglyph ! start in stereoscopic mode HYPERGEO Version 2.1 Page C-1 Appendix C. Command Line Arguments The HYPERGEO command line consists of the name of the program's executable file, HYPERGEO.EXE, followed optionally by additional parameters that are passed to the program. This additional text is used to transmit control specifications to the HYPERGEO program. Parameters passed to a program in this manner are called command line arguments. Note: Normally the .EXE extension is not typed when the program is being executed since DOS knows it without being told. The HYPERGEO program accepts a number of command line arguments that can be used to control the behavior of the program. Many of the command line arguments perform a function identical to parameters that can be specified in the HYPERGEO configuration file. Whenever a command line argument and a configuration file parameter are in conflict, the command line argument takes precedence. Furthermore, the command line specifications will continue to override configuration parameters that might be contained in any new geometry definition files that are read into the program interactively. Each command line argument consists of a single key letter that is entered on the HYPERGEO command line. The key letters should be typed on the command line with a preceding hyphen, for example, HYPERGEO -a Multiple arguments can either be strung together following a single hyphen or typed separately preceded by individual hyphens; thus, HYPERGEO -aIv and HYPERGEO -a -I -v are equivalent. Several arguments are flags indicating that a file name follows them immediately on the command line. For these arguments, there must be no other argument letters following them before the file name. There may, however, be one or more (or no) spaces between the flag letter and the file name. HYPERGEO Version 2.1 Page C-2 In all instances, the case (lower or upper) of the key letter is significant. A number of arguments use the two cases of the same letter to mean opposite settings of a single parameter. All arguments that use only one case of a letter require it to be lower-case; it will be rejected as an error if typed in upper-case. (Confession: The standard notation of using a hyphen to prefix command line argument letters is styled after the Unix convention. HYPERGEO follows this usage, and hyphens will always work. However, to be candid, the arguments will also be handled correctly if preceded by a slash ('/', NOT a back-slash) or by nothing at all. The only possible source of confusion is with file names appearing in the command line, but as long as all file names are always correctly and immediately preceded by their flag argument, the hyphen (or slash) prefix is superfluous and can be dispensed with.) Most arguments perform an independent function and can be used in any desired combination with other arguments. However, five of the arguments serve to set the value of a single parameter, the 3-D graphic display mode, and they are mutually exclusive. These arguments are -a, -h, -p, -r, and -s; if more than one of them appears in a HYPERGEO command line, the one typed last takes effect. Also, the -e and -x arguments specify alternative modes of overlay swapping; if both are entered, the last one typed takes effect. The remainder of this appendix contains a list of all command line arguments for the HYPERGEO program. HYPERGEO Version 2.1 Page C-3 Argument: -a Use: Selects the stereoscopic anaglyph mode of display. This argument is mutually exclusive with -h, -p, -r, and -s. Default: Perspective display mode. Argument: -b Use: Instructs HYPERGEO to create all graphics in monochrome, using black on a white background, even if the graphics display device supports color. This can be useful when a monochrome monitor is connected to a color graphics controller card. With monochrome graphics the stereoscopic anaglyph and surface relief display modes are unavailable, and solid surface fill always uses patterns. Default: Color if graphics device supports it, else monochrome. Argument: -c Use: Specifies the name of a HYPERGEO configuration file to use. The argument letter must be followed on the command line by the file name, with no other arguments intervening. The file name can contain full DOS path information; if there is no path specified, it is assumed to exist in the current directory. If the file name specification has no extension, it is assumed to have the extension .CFG. If this argument is used and the specified file cannot be opened, it is a fatal error and HYPERGEO terminates. It is not an error, however, if the default configuration file, HYPERGEO.CFG, cannot be opened. Default: HYPERGEO.CFG in current directory HYPERGEO Version 2.1 Page C-4 Argument: -d/-D Use: Specifies whether or not to make use of the system's installed mouse driver. The lower-case option, -d, causes HYPERGEO to use the mouse driver. The upper-case option, -D, tells the program to use its own mouse support functions without making use of the system's driver. These options should rarely be used. They are provided only for those exceptional cases where a particular mouse driver fails to operate correctly with HYPERGEO. Almost all standard mouse drivers will provide proper functioning of the mouse. However, a few drivers have been encountered that produce erratic mouse behavior in HYPERGEO. If you are experiencing less than perfect mouse performance - for example, occassional mouse button presses or releases that fail to have an effect - try using the -D option to bypass the mouse driver. Note: The lower-case option, -d, is the default setting and thus has essentially no effect; it is included merely for completeness and for possible future extensions. Default: Use the system's installed mouse driver. Argument: -e Use: Tells HYPERGEO to use expanded (EMS) memory as its overlay swapping area. Enabling overlay swapping can speed up program execution. To use this option, your system must either have an EMS-standard expanded memory board installed or be set up for EMS emulation with a driver such as Microsoft's EMM386.EXE. This option is mutually exclusive with the -x option. If both are specified, the last one entered takes effect. Note: Don't enable overlay swapping if your system already has a disk caching program installed. Default: Overlay swapping is disabled. HYPERGEO Version 2.1 Page C-5 Argument: -f Use: Specifies the name of a HYPERGEO geometry definition file to use. The argument letter must be followed on the command line by the file name, with no other arguments intervening. The file name can contain full DOS path information; if there is no path specified, it is assumed to exist in the current directory. If the file name specification has no extension, it is assumed to have the extension .BIN. If either the geometry file specified by this argument or the default geometry file cannot be opened, it is a fatal error and HYPERGEO terminates. Default: HYPRCUBE.BIN in current directory This is the standard four-dimensional hypercube. Argument: -g/-G/-gG Use: Selects the method for reducing the 4-D geometry to a 3-D image for display. The lower-case argument, -g, selects intersection mode, and the upper-case argument, -G, selects projection mode. If both the lower-case and upper-case arguments are specified (i.e., -gG), a dual 4-D to 3-D reduction is performed; this generates both a projection image and an intersection image displayed simultaneously side by side on the screen. Default: Intersection mode Argument: -h Use: Selects the hidden-line mode of 3-D display. This argument is mutually exclusive with -a, -p, -r, and -s. Also, hidden-line display mode cannot be used with 4-D projections. If this argument is used together with the -G argument, a warning message is issued and the program starts up in perspective display mode. Default: Perspective display mode. HYPERGEO Version 2.1 Page C-6 Argument: -i/-I Use: Specifies whether or not to display the special HYPERGEO information window automatically upon program start up and whenever a new geometry definition file is read. The lower-case argument, -i, selects automatic display, and the upper-case argument, -I, deactivates it. Default: The information window is automatically displayed. Argument: -m/-M Use: Specifies whether or not to display the menu area along the right edge of the screen. The menu area lists a number of parameters that describe the current display and that can be modified interactively within the HYPERGEO program. It also contains a mouse control pad area which allows the mouse to be used for performing rotations of the geometry. If the menu area is not displayed, the primary geometry graphics use the entire screen area; the mouse may not be used without the display of the menu area. The lower-case argument, -m, selects display of the menu area, while the upper-case argument, -M, causes the menu not to be displayed. Default: The menu area is displayed. Argument: -n/-N Use: Specifies whether or not to display the name of the geometry definition file in the upper, left corner of the graphics display. The lower-case argument, -n, selects display of the file name, while the upper-case argument, -N, causes the name not to be displayed. Default: The geometry definition file name is displayed. HYPERGEO Version 2.1 Page C-7 Argument: -p Use: Selects perspective display mode. This argument is mutually exclusive with -a, -h, -r, and -s. Default: Perspective display mode. Argument: -r Use: Selects the surface relief mode of 3-D display. This argument is mutually exclusive with -a, -h, -p, and -s. Surface relief mode cannot be used with 4-D projections. If this argument is used together with the -G argument, the program uses perspective display mode. Default: Perspective display mode is used. Argument: -s Use: Selects solid display mode. This argument is mutually exclusive with -a, -h, -p, and -r. Also, solid display mode cannot be used with 4-D projections. If this argument is used together with the -G argument, a warning message is issued and the program starts up in perspective display mode. Default: Perspective display mode. HYPERGEO Version 2.1 Page C-8 Argument: -v/-V Use: Specifies whether or not to display the unit axis vectors along with the primary geometry graphics. The lower-case argument, -v, selects display of the axis vectors, while the upper-case argument, -V, causes the axis vectors not to be displayed. Note: Regardless of the setting of this option, the axis vectors are never displayed in hidden-line, surface relief, or solid display mode. Default: The unit axis vectors are not displayed. Argument: -w Use: Instructs HYPERGEO to create all graphics in monochrome, using white on a black background, even if the graphics display device supports color. This can be useful when a monochrome monitor is connected to a color graphics controller card. With monochrome graphics the stereoscopic anaglyph and surface relief display modes are unavailable, and solid surface fill always uses patterns. Default: Color if graphics device supports it, else monochrome. Argument: -x Use: Tells HYPERGEO to use extended memory as its overlay swapping area. Enabling overlay swapping can speed up program execution. To use this option, your system must have extended memory (memory at addresses above 1MB), and an extended memory driver (such as Microsoft's HIMEM.SYS) must be installed. This option is mutually exclusive with the -e option. If both are specified, the last one entered takes effect. Note: Don't enable overlay swapping if your system already has a disk caching program installed. Default: Overlay swapping is disabled. HYPERGEO Version 2.1 Page C-9 Argument: -2/-3 Use: Specify the system's CPU type. The -2 option tells HYPERGEO that it is running on an 80286-based computer (or on an earlier CPU, i.e., an 8088). Such computers are the IBM PC, PC-XT, and PC-AT, and their immediate contemporaries. The -3 option tells the program that it is running on an 80386-based system (or on a later CPU, i.e., an 80486). The key distinction between these two groups of computers is that the earlier CPUs were limited to 16-bit arithmetic while the newer CPUs can perform 32-bit arithmetic. These options should almost never be used. HYPERGEO has a built-in function that checks the identity of the system's CPU when it starts up. It uses this information to determine whether or not it can execute 32-bit arithmetic commands. In almost all cases this automatic check works properly to indicate the correct CPU type. However, it is possible that some computers may not be correctly identified this way by the program. If an 80386 or 80486 CPU is misidentified as an earlier 16-bit processor, the symptoms are very subtle (just a slight loss of execution speed), and it is very unlikely that you would ever notice the difference. If an 80286 or 8088-based computer is misidentified as a 32-bit machine, the symptoms are much harder to miss: HYPERGEO will crash upon start-up. If HYPERGEO crashes or hangs the system almost immediately upon start-up and you have a computer with an 80286 or 8088 CPU, try using the -2 option. If you have an 80386 or 80486 in your system and you are the sort who tends to worry, you can go ahead and use the -3 option. It will do no harm, and it will guarantee that HYPERGEO is making proper use of your computer's 32-bit arithmetic capabilities. Note: Using the -3 option with an 80286 or 8088-based system will do harm; it will cause HYPERGEO to crash. Default: No default; setting determined by checking the CPU HYPERGEO Version 2.1 Page C-10 Argument: -? Use: Prints to the screen a description of the appropriate syntax and usage of these command line arguments and terminates the HYPERGEO program. Default: No syntax description is printed. The following is an example of a HYPERGEO command line: HYPERGEO -f 16cell -cc:\configs\special.dat -Gav -I This command invokes the following command line argument functions: * Use the geometry definition file 16CELL.BIN located in the current directory * Use the configuration file SPECIAL.DAT located in the directory C:\CONFIGS\ * Use projection mode for reducing the 4-D geometry to a 3-D image * Use stereoscopic anaglyph display mode for the 3-D image * Display the unit axis vectors * Deactivate the automatic display of the HYPERGEO information window HYPERGEO Version 2.1 Page D-1 Appendix D. Interactive Commands When the HYPERGEO program is executed, it accepts any command line arguments, reads in the appropriate configuration file (if any) and geometry definition file, and generates the initial display of the geometric object. From that point on, the program remains responsive to a set of interactive commands which enable the user to manipulate the orientation of the object and to modify the various parameters that control the manner in which the display is generated. Many of these interactive commands affect the same parameters that can be initially set through the configuration file and command line arguments. All interactive commands are invoked using either the keyboard or the mouse. The mouse (if the system has one) provides an alternative means for invoking the commands for rotation of the object and for changing the value of the several display parameters that are listed in the menu area; if there is no mouse, keyboard commands can be used to perform the same functions. (Also, if the menu area is not displayed, the mouse cannot be used, and the keyboard equivalents must be similarly resorted to instead.) D.1 Keyboard Commands and the Modifier Keys The keyboard commands consist of either a single keystroke or a single keystroke accompanied by the simultaneous pressing of one of the standard keyboard modifier keys: Shift, Ctrl, or Alt. These modifier keys are only used with some of the interactive commands, and in almost all cases they have a consistent meaning: Shift - Generally used to reverse the sense or direction of the command's action. Used with keystrokes that invoke the various rotations, it causes the direction of rotation to be reversed, that is, to be equivalent to a negative turn angle. Note: For all command keystrokes that are letters, if the Shift key has no special meaning for that command, the command may be entered equivalently in either lower or upper case. Only four letter commands (G, X, Y, and Z) have special Shift-modified meanings. HYPERGEO Version 2.1 Page D-2 Ctrl - Used with keystrokes for the rotation commands plus the command for translating the intersecting hyperplane along the W-axis in hyperspace; causes the command action to be repeated continuously as fast as the computer's processing speed will allow. This "command automation" is stopped by pressing any key. In the case of the hyperplane translation command, the automated motion is automatically reversed in direction each time the intersecting hyperplane reaches an extreme of the hyperobject. This causes the hyperplane to move continuously back and forth through the entire body of the hyperobject. Alt - Used with keystrokes for the rotation commands; has the same "command automation" effect as the Ctrl modifier except the rotation is in the reverse direction. The continuous, automated rotation is stopped by pressing any key. D.2 Mouse Selected Interactive Commands When the mouse is used to invoke a rotation command, no special modifiers are necessary since the action of the mouse buttons provides the equivalent control. A rotation command can be invoked with the mouse (whenever the HYPERGEO menu area is displayed) by moving the mouse so the cursor is over the selection box labeled with the coordinate letters indicating the desired axis or plane of rotation; this will highlight the selection box. To begin the rotation, press the left or right mouse button; the right mouse button causes a rotation in the positive direction, and the left mouse button rotates the object in the negative direction. The rotation will continue as long as the mouse button remains pressed. A single incremental rotation can be achieved by quickly clicking and releasing the mouse button. There are nine rotation command selection boxes in the HYPERGEO menu area, divided into two sections: 3-D axial rotations, and 4-D planar rotations. All nine are available for 4-D geometries, but only the three 3-D rotations can be selected when the input geometry is three-dimensional (in 3-D geometry mode, the section of the mouse control pad area that contains the selection boxes for 4-D rotations is shaded over with a stippled pattern, and those mouse functions cannot be selected). The 4-D rotation selection boxes are labeled with the letters of the two coordinates specifying the plane of the rotation; thus, selecting the box labeled "XY" initiates a rotation in hyperspace of the XY-plane. The 3-D rotation selection boxes are labeled with the single letter of the axis around which the rotation occurs: X, Y, or Z. HYPERGEO Version 2.1 Page D-3 If your system has no mouse or if the menu area isn't displayed (or if you simply prefer to use the keyboard), there is an equivalent keyboard command for each of the nine mouse rotations: Keystroke Mouse Selection Box Label --------- ------------------------- X X Y Y Z Z F5 YZ F6 XW F7 XZ F8 YW F9 XY F10 ZW The keystrokes F5, F7, and F9 correspond to the YZ, XZ, and XY 4-D planar rotations. Note that these are the three rotations that occupy the top row of the six 4-D rotation selection boxes in the menu area and that they are the 4-D rotations that don't involve the W coordinate. These three rotations produce the same visual effect in the geometry display as the three 3-D rotations whose selection boxes are directly above them in the menu: X, Y, and Z. The keystrokes F6, F8, and F10 correspond to the three 4-D rotations that do involve the W coordinate: the XW, YW, and ZW planar rotations. These are the 4-D rotations that occupy the bottom row of selection boxes within the menu area. They do not correspond in effect to any of the 3-D rotations. When the alternative user interface state that allows the temporary translation of a selected 4-D vertex is in effect, all 4-D rotations are disabled; that portion of the menu is replaced by a control pad area that has selection boxes for executing translations of the vertex along each of the four coordinate axes. These boxes are labeled X, Y, Z, and W. To perform a translation, move the mouse cursor so the appropriate box is highlighted. Pressing the right mouse button moves the vertex in the positive direction along the particular axis, and pressing the left mouse button moves it in the negative direction. The vertex continues to move until the mouse button is released. In addition to performing interactive rotations and vertex translations, the mouse can be used to modify the other parameters shown in the menu area. To do this, move the mouse cursor over the label of the desired parameter and click one of the buttons. Depending on the particular parameter, this will either cause an immediate change to the parameter or bring up a dialog box requesting specification of a new value. HYPERGEO Version 2.1 Page D-4 The parameters for eye-to-screen distance (EYE DIST), scale factor (SCALE), W-coordinate of the intersecting hyperplane (W-INTER), and the W-coordinate of the 4-D projection viewpoint (W-PROJ) exhibit the first type of behavior; selecting them with the mouse results in an immediate change to the parameter's value. This action is equivalent to the behavior of the keystroke commands that are used to modify these same parameters. Which mouse button is used to select the parameter determines the direction in which its value is adjusted: the right button increases it, and the left button decreases it: EYE DIST - Selecting with the right mouse button increases the eye-to-screen distance by 10% (thus decreasing the perceived degree of perspective). Using the left mouse button decreases the eye-to-screen distance by 10%. SCALE - Selection with the right mouse button increases the scale factor by 10%. Using the left mouse button decreases the scale factor by 10%. W-INTER - Selecting this parameter with the right mouse button increases the W-coordinate value of the intersecting hyperplane by an amount equal to the current value of the delta-W parameter. Selection with the left mouse button decreases the W-coordinate value by the delta-W amount. W-PROJ - Selecting with the right mouse button increases the W-coordinate of the 4-D projection viewpoint by 10% (thus decreasing the perceived degree of perspective). Using the left mouse button decreases the coordinate by 10%. All other parameters shown in the menu are modified through a dialog box that pops up when their label area is selected with the mouse (either button). When the geometry being displayed is three-dimensional, the parameters relating to the translation of the four-dimensional intersecting hyperplane (W-INTER) and the position of the 4-D projection viewpoint (W-PROJ) are not displayed since they have no meaning in 3-space. HYPERGEO Version 2.1 Page D-5 D.3 Interactive Parameter Specification Some of the interactive commands prompt for the input of a value to be assigned to a program parameter. This value is generally either a numeric quantity, a file name, or a particular key word. The input is requested from the user via a prompting dialog box which pops up in the primary graphics area when the command is executed. There are several different types of dialog box used in HYPERGEO depending upon the nature of the parameter to be assigned a value. The behavior of these has been described fully in an earlier section of this document (3.2.3). Of these, the most frequently used is the generic text input dialog box. A text input dialog box contains an entry line in which the user's typed response appears. After typing in the appropriate value, the user should indicate that the entry is complete and correct by selecting the "Ok" button with the mouse (or by pressing <Enter>); this will cause the command to accept the typed in value for the given parameter. A dialog box may be cancelled at any time by selecting the "Cancel" button with the mouse (or by pressing <Esc>); this aborts the command that caused the dialog box to appear. Dialog boxes that accept character input - either key words or file names - are case insensitive; upper or lower case may be used as desired. Numeric values may be either integers or floating-point numbers depending upon the context of the command. If a floating-point number is appropriate, it may be entered with an optional decimal point and fractional digits and may include an optional decimal exponent in scientific notation. If the user's entry in a prompting dialog box is invalid, an error message window will appear. Like all HYPERGEO message windows, it can be dismissed by selecting the "Ok" button with the mouse (or by pressing <Enter> or <Esc>). All errors resulting from incorrect input to interactive parameter prompts are non-fatal; the program will continue running, ready for the next interactive command, once the message window is dismissed. HYPERGEO Version 2.1 Page D-6 D.4 HYPERGEO Interactive Keyboard Commands The following is a list of all HYPERGEO interactive commands that can be entered from the keyboard while the program is displaying an active geometry: Keystroke: A Use: Selects the stereoscopic anaglyph mode of display. If the program is operating with monochrome graphics, the stereoscopic anaglyph display cannot be used; a message window will appear indicating this, and the current display mode will not be changed. Keystroke: B Use: Toggles on and off the color spectrum banding of 4-D graphics according to the W-coordinate of the corresponding vertexes and edges in the current hyperobject. The overall range of W-coordinate values is divided into ten sub-divisions, and each is assigned a different color from a pre-programmed spectrum of ten colors. These start with blue-violet for the minimum W-coordinate, proceed through blue, aqua, cyan, blue-green, green, olive-green, yellow, and orange for progressively larger W values, and end with red for the maximum W-coordinate. Each edge - or sub-segment of an edge - is displayed in the color corresponding to the spectrum band of its W-coordinate. If the program is operating with monochrome graphics, the color spectrum banding feature cannot be used. HYPERGEO Version 2.1 Page D-7 Keystroke: C Use: Accepts a new color specification for drawing the primary graphics. For perspective and hidden-line display modes, this will be the single EGA color used to draw the edge outlines of the geometrical image being displayed. (This same primary graphics color is also used for the transient data value fields within the HYPERGEO menu area and for text appearing in dialog box windows.) The command will display a list dialog box offering a choice of one of the sixteen standard EGA colors: BLACK DARKGRAY BLUE LIGHTBLUE GREEN LIGHTGREEN CYAN LIGHTCYAN RED LIGHTRED MAGENTA LIGHTMAGENTA BROWN YELLOW LIGHTGRAY WHITE Note: The current background color cannot be selected as the primary foreground color (for obvious reasons); its entry will be disabled in the list dialog box. For solid display mode, if the system has VGA color capability, the command will accept a new specification for the twelve graduated shades of color to be used for solid surface rendering. If the system has a mouse, a color selection dialog box will appear which can be used to preview and choose the precise shade desired. Otherwise, a text dialog box will prompt for a color specification in the form of a red/green/blue number triple - three integers in the range 0 to 63 which are the relative intensities of the three primary components. In either instance, the color selected defines the lightest of the twelve solid-fill shades; the program will calculate eleven additional, darker shades from this to create the full range required for realistic surface rendering. This command is not applicable to the display modes that use red/blue anaglyphs (i.e., stereoscopic and surface relief) since they use colors that are fixed by the requirements of the "3D" viewing glasses. HYPERGEO Version 2.1 Page D-8 Keystroke: D Use: Accepts a new delta value that will be used to increment (or decrement) the W-value of the intersecting hyperplane. This delta value is used by the interactive command which performs a translation of the intersecting hyperplane on geometries being displayed via the 4-D intersection mode (see Page Up/Page Down below). The value entered is a positive floating-point number. It is in the units of the currently active geometry. Keystroke: E Use: Accepts a new value for the eye-to-screen distance. This parameter is used in the generation of all 3-D to 2-D screen projections. Decreasing the eye-to-screen distance enhances the amount of apparent perspective in the display. For greatest realism, the eye-to-screen distance should be set to correspond to the true distance between the viewer's eyes and the screen. HYPERGEO automatically limits the maximum and minimum values to which this parameter can be set. This prevents numeric errors from occurring during the program's computations. The value entered is a positive floating-point number. It is in inches. Keystroke: F Use: Accepts the name of a new geometry definition file. A file selection dialog box appears which lists all files with the default extension for binary geometry files (.BIN) in the current directory. The features of the dialog box may be used to select valid (i.e., binary) geometry definition files in other directories and with non-standard extensions. Reading in a new geometry definition file completely reinitializes HYPERGEO. The current geometry's database is freed, its display is cleared, and any configuration information (either from the command line or from a configuration file) is reapplied. HYPERGEO Version 2.1 Page D-9 Keystroke: G Use: Cycles among the three variations of 4-D to 3-D geometry reduction: intersection, projection, and dual (simultaneous intersection and projection). The case of the entered keystroke is significant. Lower case 'g' cycles in the order Intersection/Projection/Dual; upper case 'G' in the order Dual/Projection/Intersection. Because hidden-line, surface relief, and solid displays cannot be used with 4-D to 3-D projections, if one of those three modes is in effect when the program switches from 4-D intersection to either 4-D projection or to dual intersection/projection, perspective display mode is automatically selected. The G command is not applicable to three-dimensional geometric objects. Keystroke: H Use: Selects hidden-line display mode. If the current geometrical object is four-dimensional and the 4-D to 3-D reduction mode is projection or dual intersection/projection, the hidden-line display cannot be used. HYPERGEO Version 2.1 Page D-10 Keystroke: I Use: Displays the special HYPERGEO Information Window. This window contains a summary of the current object's topology, plus a number of related program parameters. For the topology, the window shows: * number of dimensions * number of vertex points * number of edges * number of faces * number of hyperfaces (4-D geometries only) * span (geometry units) * radius (geometry units) The "span" of an object is calculated by finding the minimum and maximum extents of the object in each of its three or four dimensions; the span is the largest difference between the minimum and maximum extents in a single dimension. The "radius" of an object is the greatest distance in (hyper)space of any vertex point from the origin. Additional parameters shown are: * screen height and width (inches) * horizontal and vertical pixels per inch * free memory (bytes) The HYPERGEO Information Window is an ordinary message dialog box; it can be dismissed by selecting the "Ok" button with the mouse (or by typing <Enter> or <Esc>). HYPERGEO Version 2.1 Page D-11 Keystroke: J Use: Replays a previously recorded journal file. A file selection dialog box appears which lists all files in the current directory that have the default journal file extension (.JNL). The features of the dialog box can be used to select for replay a journal file in a different directory or with a non-standard extension. Once a journal file name has been selected and confirmed, HYPERGEO enters the special journal replay command mode. This is described fully in Section 3.6.2 of this document. Keystroke: K Use: Toggles on and off the recording of a journal file. A dialog box is used to accept the name of a new journal file to record. If a file already exists with the specified name, a second dialog box appears requesting approval to go ahead and overwrite it. While a journal file is being recorded, a message is displayed at the bottom of the display screen so indicating. HYPERGEO Version 2.1 Page D-12 Keystroke: L Use: Locates the geometrical model at a particular orientation or position. This command has two variations depending upon the state of the user interface. In the default state in which the interface presents a full command set for performing rotations of the geometry in space or hyperspace, the Locate command accepts three (for 3-D) or four (for 4-D) angles which specify the desired angular displacements of the X, Y, Z, and W-coordinate axes from their initial orientations. Note that these are the same angular displacements as are shown in the menu area under the AXIS ANG label. A text dialog box is used to accept the angle values; they should be entered as three or four positive real values in the range of 0 to 180 degrees. (Note: Due to limitations of the algorithm used to calculate the rotations necessary to achieve the specified axis angles, for 4-D geometries one of the four angles must be zero degrees. This limitiation does not exist for 3-D geometries. Also, for both 3-D and 4-D not all combinations of angles represent physically possible orientations.) In the alternative state in which the user interface permits the temporary translation of a selected 4-D vertex, the Locate command accepts four real values that are the X, Y, Z, and W-coordinate values in geometry units of the desired position in 4-space of the selected vertex. This allows the vertex to be located precisely at any valid coordinate within the range of the command. When the user interface is in the vertex translation state and a vertex has been selected, the coordinates are displayed in the menu area under the VERTEX label. (Note: In order to avoid numeric computation errors, the program will not accept any coordinate value which has an absolute value greater than 1.25 times the radius of the current geometry.) HYPERGEO Version 2.1 Page D-13 Keystroke: M Use: Toggles on and off the display of the HYPERGEO menu area. The menu area must be displayed to enable the use of the mouse for command selection. Keystroke: N Use: Toggles on and off the display of the name of the current geometry definition file. When its display is toggled on, the geometry file name is shown in the upper-left of the main graphics area. Keystroke: O Use: Outputs the current image of the entire screen into a file in PCX graphics format. A dialog box prompts for the file name. If a file already exists with the specified name, a second dialog appears requesting approval to go ahead and overwrite it. Keystroke: P Use: Selects perspective display mode. Keystroke: Q Use: Quits the HYPERGEO program and returns to DOS. Keystroke: R Use: Selects surface relief display mode. If the current geometrical object is four-dimensional and the 4-D to 3-D reduction mode is projection or dual intersection/projection, the surface relief display mode cannot be used. HYPERGEO Version 2.1 Page D-14 Keystroke: S Use: Selects solid display mode. If the current geometrical object is four-dimensional and the 4-D to 3-D reduction mode is projection or dual intersection/projection, the solid display mode cannot be used. Keystroke: T Use: Accepts a new value for the turn angle. The turn angle is the basic incremental angle used by each of the rotation commands. A large turn angle is useful for rapid manipulation of the geometrical object under study. Smaller angles can be used for fine positioning. The value entered is a positive floating-point number. Its units are degrees. Keystroke: U Use: Toggles between the two states of the user interface. In the default state, the user interface presents a full set of commands for performing rotations of the geometry in three and four dimensions. In the alternative state, a particular vertex may be selected in a 4-D geometry and may be temporarily repositioned in hyperspace. In this state, the 4-D rotation commands in the menu are replaced by the commands for performing vertex translation. Keystroke: V Use: Toggles on and off the display of the unit axis vectors. Regardless of the setting of this toggle, the unit axis vectors are not displayed with hidden-line, surface relief, or solid display mode. HYPERGEO Version 2.1 Page D-15 Keystroke: W Use: The function of this command depends on whether the current 4-D image is an intersection or a projection. For 4-D to 3-D intersection images, it accepts a new value for the W-coordinate of the hyperplane used to calculate the intersection. Repositioning the intersecting hyperplane at various W-coordinate values provides a means for examining the hypergeometrical structure of the current object. For 4-D to 3-D projection images, the 'W' keystroke accepts a new value for the W-coordinate of the theoretical viewpoint used to calculate the projection. The viewpoint is always located on the positive W-axis; changing its distance from the hyperobject's origin adjusts the degree of perspective visible in the projection. For both usages of the command the value entered is a floating-point number in geometry units. Note: In the dual imaging mode in which an intersection and a projection both are displayed at the same time, the 'W' command affects the intersecting hyperplane, not the projection viewpoint; thus, it has no effect on the appearance of the projection image. HYPERGEO Version 2.1 Page D-16 Keystroke: X Y Z Use: Executes a three-dimensional rotation of the current 3-D image about the particular axis: X, Y, or Z. The size of the rotation is determined by the current turn angle. When the active geometry is four-dimensional, all 3-D rotations are transient and operate only on the immediate 3-D image which has been derived from the hyperobject's geometry via a 4-D to 3-D reduction using either projection or intersection. In this case, the program does not update the display of the unit axis vectors or the values shown in the menu's axis angle data fields during 3-D rotations. For three-dimensional geometries, however, 3-D rotations are permanent; the internal database is modified accordingly, and the unit axis vectors and the displayed values for the axis angles are updated with each 3-D rotation. By themselves, the X, Y, and Z keys produce a single incremental rotation in the positive direction. This effect can be altered by combining the keystroke with the simultaneous pressing of one of the standard keyboard modifier keys (for example, Shift-X is entered by pressing and holding down the Shift key, then pressing X): Shift - Reverses the direction of the rotation Ctrl - Automates continuous rotations in the positive direction. The rotations are performed as fast as the system's processing speed will allow. To stop the animation, press any key. Alt - Automates continuous rotations in the negative direction. The rotations are performed as fast as the system's processing speed will allow. To stop the animation, press any key. HYPERGEO Version 2.1 Page D-17 Keystroke: F1 Use: Displays the main on-line help screen. This screen contains an overview of the HYPERGEO program and a menu listing additional help pages that are organized by command categories. Each contains a brief description of a set of interactive commands available within the HYPERGEO program that perform related functions. The on-line help facility is described more fully in Section 3.2.2 of this document. The on-line help screen can be dismissed by typing <Esc>. This returns the program to the graphics state it was in prior to the F1 command. Keystroke: F2 Use: Pauses the HYPERGEO program and brings up a temporary DOS command screen. This permits the user to execute DOS commands without completely exiting HYPERGEO. Typing EXIT at the DOS prompt closes the DOS screen and returns to HYPERGEO; the program's status is restored to that existing prior to executing the F2 command. A temporary DOS command screen which is invoked from within a running program like this is commonly referred to as a "DOS shell". Note: While the temporary DOS command screen is active, the HYPERGEO program is still occupying its share of the computer's memory. Since this leaves considerably less free memory than is normally available, some DOS commands may be unable to run in the F2 DOS shell. Keystroke: F3 F4 Use: Performs continuous automated random rotations. The keystroke F3 initiates the continuous execution of the three 3-D rotations in a randomly determined order, and F4 initiates random execution of the six 4-D rotations. All rotations are in a positive angular direction with a size determined by the current turn angle. To stop the automation, press any key. HYPERGEO Version 2.1 Page D-18 Keystroke: F5 F6 F7 F8 F9 F10 Use: Executes a four-dimensional rotation of the current hyperobject's geometry in hyperspace. The six function keys F5 through F10 correspond to the six possible coordinate planes in which a 4-D rotation may occur: F5 - YZ plane F6 - XW plane F7 - XZ plane F8 - YW plane F9 - XY plane F10 - ZW plane The size of the rotation is determined by the curent turn angle. After each 4-D rotation, the hyperobject's geometry is reduced to a 3-D image using the currently active method (intersection or projection), and the 3-D image is converted into a 2-D display on the computer's screen using the currently active display mode (perspective, stereoscopic anaglyph, hidden-line, surface relief, or solid). By themselves, the F5 through F10 keys produce a single incremental rotation in the positive direction. This effect can be altered by combining the keystroke with the simultaneous pressing of one of the standard keyboard modifier keys (for example, Shift-F5 is entered by pressing and holding down the Shift key, then pressing F5): Shift - Reverses the direction of the rotation Ctrl - Automates continuous rotations in the positive direction. The rotations are performed as fast as the system's processing speed will allow. To stop the animation, press any key. Alt - Automates continuous rotations in the negative direction. The rotations are performed as fast as the system's processing speed will allow. To stop the animation, press any key. HYPERGEO Version 2.1 Page D-19 Keystroke: 1 2 3 4 5 6 Alt-1 Alt-2 Alt-3 Alt-4 Alt-5 Alt-6 Note: Only the commands '1', '2', and '3' are available for 3-D geometries. Use: Rotates the object to one of the possible distinct orthogonal orientations. An "orthogonal orientation" is one in which all coordinate axes have been rotated an exact multiple of 90 degrees from their original positions. The orientations generated by these commands all have the property that a set of three of the axes are positioned coincident with the original positions of the X, Y, and Z axes. This means that the three axes form a normal "left-handed" 3-D coordinate system, that is, one in which the positive Z-axis is directed away from the viewer (into the screen) when the positive X-axis points to the right and the positive Y-axis points up. For 3-D geometries there are three such orthogonal orientations, and for 4-D there are twelve. For 4-D objects the fourth coordinate axis is rotated into the position of the original W-axis. This is the coordinate axis along which the intersecting hyperplane and the projection viewpoint are positioned for 4-D to 3-D image reduction. The orientations can be identified by specifying which axes are rotated into each of the original unrotated axis positions. For example, the orientation designated YZXW has the Y-axis positioned along the original X-axis, the Z-axis along the original Y-axis, the X-axis along the original Z-axis, and the W-axis still in its original unrotated direction. HYPERGEO Version 2.1 Page D-20 The twelve commands generate the following orientations: Orientation Keystroke 3-D 4-D --------- ------------------- 1 XYZ XYZW 2 YZX YZXW 3 ZXY ZXYW 4 YWZX 5 XWYZ 6 ZWXY Alt-1 YXWZ Alt-2 ZYWX Alt-3 XZWY Alt-4 WYXZ Alt-5 WXZY Alt-6 WZYX Notice that the Alt-modified version of each 4-D command differs from the unmodified numeric key version by having the same two axes form the rotated XY-plane (but in reversed order) and the same two axes form the ZW-plane (but also in reverse order). Compare, for example, the rotation produced by the '6' command (ZWXY) with that of the Alt-6 command (WZYX); the pairs ZW/WZ and XY/YX are reversed between the two. Note: The orientation produced by the '1' command (XYZ for 3-D or XYZW for 4-D) is the unrotated orientation of the original geometry. This is the same as the orientation generated by the Home command. However, unlike the Home command, the '1' command does not save the current orientation for restoring with the End command. Note: Use the numeric keys on the top row of the regular keyboard for these commands, not the keypad numeric keys. Keystroke: 0 Use: Positions the display of the graphical image on the screen so that the point corresponding to the origin of the geometrical model is at the center of the main display area. The '5' key in the keypad (on keyboards with keypads and with NumLock off) also performs this function. HYPERGEO Version 2.1 Page D-21 Keystroke: Home Use: Restores the orientation of the current geometry as it existed when the geometry definition file was first read in. All axis angles are reset to zero degrees. The current orientation is saved and may be returned to by using the End command (see below). Keystroke: End Use: Restores the current geometry to the orientation it was in prior to the most recent Home command (see above). If no Home command has yet been performed on the current geometry, the End command has no effect. Keystroke: = Use: Accepts a new value for the current display scale factor. A larger scale factor increases the size of the image on the screen, while a smaller scale factor decreases it. The value entered is a positive floating-point number. Its units are screen inches per geometry unit. Keystroke: + - Use: Performs a "zoom" operation on the visible image. The '+' command enlarges the graphical display, and the '-' command reduces it. These commands operate by changing the display scale factor. The '+' command increases the scale factor by 10% of its current value, and the '-' command decreases it by 10%. HYPERGEO Version 2.1 Page D-22 Keystroke: Tab Use: Toggles on and off the highlighting of all edge lines in the displayed image that belong to a particular hyperface in the original 4-D geometry. The <Space> bar and <Backspace> key are used to select which hyperface is highlighted (see below). For 4-D to 3-D projection images, the highlighted edges correspond exactly to edges in the original hyperobject (they are in fact projections onto 3-space of those edges). For 4-D to 3-D intersection images, the highlighted edges form the outline of the polygonal face that is the intersection of the highlighted hyperface with the intersecting hyperplane. For a given orientation of the hyperobject and position of the intersecting hyperplane it is possible that the highlighted hyperface will not be intersected; when that is the case, no highlighting will show up in the 3-D image even when highlighting is toggled on. Hyperface highlighting may be used with dual intersection/projection images; the highlighting will appear in both images. In fact, this can be a very revealing technique for visualizing the relation between 4-D projection and intersection images. Note: Hyperface highlighting does not appear if an intersection image is drawn in hidden-line, surface relief, or solid display mode. Keystroke: Space Backspace Use: Cycles through all hyperfaces in a 4-D hyperobject for highlighting (see <tab> command above). The <Space> bar cycles in a forward direction (per an arbitrary ordering), and the <Backspace> key cycles in the reverse direction. If hyperface highlighting is toggled on, the results of stepping forward or back to a different hyperface are immediately reflected in the display. If highlighting is toggled off, there is no change in the current display, but the hyperface cycling is still performed; the next time highlighting is toggled on, the newly selected hyperface will be the one highlighted. HYPERGEO Version 2.1 Page D-23 Keystroke: Insert Delete Ctrl-Insert Ctrl-Delete Use: Alters the degree of perceived 3-D perspective for the currently displayed image. These commands operate by modifying the eye-to-screen distance. The Insert command increases the eye-to-screen distance by 10% of its current value; this reduces the apparent perspective. The Delete command increases the perspective effect; it moves the viewpoint toward the object by decreasing the eye-to-screen distance by 10%. HYPERGEO automatically limits the maximum and minimum values to which the eye-to-screen distance can be set. This prevents numeric errors from occurring in the program's computations. When combined with the Ctrl key, Insert and Delete serve to set the eye-to-screen distance to its maximum or minimum permissable value respectively. The maximum eye-to-screen distance is useful in particular to approximate a pure orthographic projection. Keystroke: Page Up Page Down Ctrl-Page Up Ctrl-Page Down Use: The function of the Page Up and Page Down commands depends on whether the current image is a 4-D projection or intersection. For a 4-D to 3-D intersection image, the command performs a translation of the intersecting hyperplane. The hyperplane is moved along the W-axis in hyperspace. The size of the translation is determined by the value of the delta-W parameter. The Page Up command increments the W coordinate of the intersecting hyperplane by this delta amount, and the Page Down command decrements it. HYPERGEO Version 2.1 Page D-24 For 4-D to 3-D projection images, Page Up and Page Down adjust the W-coordinate of the viewpoint used to calculate the projection. The Page Up command increases the distance of the viewpoint from the hyperobject's origin by 10% of its current value; this reduces the amount of perspective seen in the image. Page Down decreases the distance of the viewpoint from the origin, enhancing the apparent perspective. By themselves, the Page Up and Page Down commands perform a single incremental action, adjusting up or down the W-coordinate value of either the intersecting hyperplane or the projection viewpoint. When combined with the simultaneous pressing of the Ctrl key (for example, to enter Ctrl-Page Up, first press and hold down the Ctrl key, then press Page Up), these keystrokes perform an enhanced variation of their basic function. For 4-D to 3-D intersections, they initiate the continuously automated motion of the hyperplane through the body of the current four-dimensional geometry. In automated mode, the direction of the hyperplane's motion is reversed each time it reaches an extreme edge of the hyperobject, that is, the point at which it ceases to intersect the hyperobject in hyperspace. Thus, the hyperplane continues to move back and forth through the entire body of the hyperobject. The automated motion is stopped by pressing any key. For 4-D to 3-D projections, Ctrl-Page Up and Ctrl-Page Down set the W-coordinate of the projection viewpoint to its maximum and minimum permissable values respectively. When the distance of the projection viewpoint from the origin is maximized, the generated 3-D image approximates a pure orthographic projection. Minimizing the viewpoint's distance creates a 3-D image with a high degree of perceived perspective. When HYPERGEO is displaying dual images of an intersection and a projection together, the Page Up and Page Down commands perform their functions for the intersection image only. That is, they affect the intersecting hyperplane (and hence the intersection image), and do not affect the projection viewpoint (or the projection image). HYPERGEO Version 2.1 Page D-25 Keystroke: Arrow Keys: Left, Right, Up, Down Use: The function of these keys depends on the state of the user interface. In the default user interface state for performing rotations of the geometry, these keys move the position of the current graphical display on the screen. When a new geometry is read in, HYPERGEO automatically centers the display within the primary graphics area. The arrow keys may be used to reposition it as required by subsequent rotations or other interactive manipulations. When the user interface is in its alternative state which permits the temporary translation of a particular 4-D vertex, the arrow keys are used to move a selected vertex in each of the four 4-D coordinate directions: Right arrow - Moves vertex in X direction Up arrow - Moves vertex in Y direction Left arrow - Moves vertex in Z direction Down arrow - Moves vertex in W direction By itself, each arrow key moves the vertex in the positive direction, i.e., increasing the particular coordinate. Combining the arrow key with the <Shift> key reverses the direction of movement; the coordinate value is decreased. (Note: Since the <Shift> key does not work consistently with the arrow keys on all keyboards, the <Ctrl> key may also may be used with the arrow keys to reverse the direction of vertex translation.) HYPERGEO Version 2.1 Page E-1 Appendix E. Running HYPERGEO under Microsoft Windows The program file supplied with HYPERGEO (HYPERGEO.EXE) is a standard DOS executable; it is not a special Windows program. However, it can be run under Windows using that system's ability to execute non-Windows DOS programs. Two files are supplied with HYPERGEO to assist in adding the program to the Windows Program Manager: an icon file (HYPERGEO.ICO), and a Program Information File (HYPERGEO.PIF). The procedure to follow to add a new program to a particular program group within the Program Manager is described in the Windows User's Guide in the section "Creating a Program Item" in the Program Manager chapter (page 78 for Windows 3.1). Within the Program Item Properties dialog box that appears, the name of the HYPERGEO PIF file (including its DOS path) should be entered in the Command Line text box. You should also supply the name of the icon file by choosing the "Change Icon..." option from the Program Item Properties dialog box. Enter the icon file name (HYPERGEO.ICO) along with its full DOS path in the File Name text box within the Select Icon dialog box. One final step may be necessary to enable the running of HYPERGEO in Windows. The supplied PIF file has an entry which specifies the program's start-up directory. This initially contains the path specification C:\HYPERGEO. If you have installed HYPERGEO in a different directory, you will have to change this entry in the PIF to reflect the actual directory in which the program executable HYPERGEO.EXE exists. To modify a PIF you use the Windows PIF Editor. This is described fully in the Section "Working with PIFs and the PIF Editor" (page 259 for Windows 3.1) and subsequent sections in the chapter "PIF Editor" in the Windows User's Guide. The PIF Editor also lets you specify any optional command line parameters you wish to pass to HYPERGEO each time it is executed. The PIF file as supplied will cause HYPERGEO to run as a full-screen DOS program. HYPERGEO Version 2.1 Page F-1 Appendix F. What's New in Version 2 Version 2 represents a major revision of the HYPERGEO program. Much of the program has been completely rewritten in an effort to achieve faster and more robust performance. In addition, many new features and enhancements to existing features have been implemented. First, a word or two about a couple of areas in which some functionality has apparently been lost from Version 1. Most notable is the fact that the program no longer supports high resolution (640 by 480 pixels) on VGA displays; EGA and VGA both use the same standard 640 by 350 resolution. This was done to make the VGA's video memory available for video page swapping (see below). It was felt that the benefits of video paging outweigh the usefulness of the higher screen resolution (unfortunately, you can't have both). Also, the Orthographic display mode no longer exists. Orthographic projections are in general the least revealing form of perspective projection. An orthographic projection can be approximated in HYPERGEO Version 2 by setting the eye-to-screen parameter to its maximum permissable value (the new Ctrl-Insert command does this). The DELTA-W parameter is gone from the menu area. This was done to make room for the new W-PROJ parameter (see below). The delta-W value can still be specified in the configuration file (the DELTA_W parameter), and can still be set interactively with the 'D' command. And lastly, the configuration file parameters COLOR_MESSAGE_TEXT and RATIO no longer exist. The primary foreground graphics color is now used for most text in all message windows (dialog boxes). The RATIO parameter has been replaced by the SCALE parameter which has a different meaning mathematically (see below). The assignment of some command keystrokes has changed. The 4-D rotation commands have been moved from function keys F1 through F6 to use keys F5 through F10. This frees up F1 for on-line help and F2 for the new DOS shell feature. The random 3-D and 4-D rotation commands have also been reassigned; they now use F3 and F4 (instead of '3' and '4'). HYPERGEO Version 2.1 Page F-2 The following is a list of new features and enhancements incorporated in HYPERGEO Version 2: * Most computations are now performed using a fixed-point data format that makes use of integer arithmetic. This provides fast execution even on systems without a math coprocessor chip (such as an 80287 or 80387). The new fixed-point computations are most efficient on 80386 (or 80486) CPUs which can perform 32-bit integer arithmetic. However, even on 80286 (or lower) systems, the performance of the program in the absence of a coprocessor has been speeded up markedly. * The hidden-line algorithm has been totally reprogrammed for faster execution. It now performs a preliminary decomposition of complex, concave objects into a set of disjoint convex shapes. These are sorted along the Z-coordinate and are displayed using the so-called "painter's algorithm". * Graphics are now drawn to two video pages which are swapped between screen updates. This produces much smoother graphics animation without the annoying "flicker" that is visible when drawing to a single video page. (Note that this necessitates abandoning 640 by 480 resolution for VGA as mentioned above.) * The portion of the program that performs the topological analysis of a new geometry definition file has been broken out into a separate stand-alone utility program, GEO2BIN. It is now only necessary to perform this analysis once each time a geometry file is created or modified rather than every time it is read into HYPERGEO. GEO2BIN stores its results in a secondary version of the geometry file that is in a binary format and can be read by HYPERGEO much more efficiently. For large and complex geometries, the time required to create a display in HYPERGEO has been reduced from minutes to seconds. * The user interface has been improved to allow additional mouse invocation of interactive commands and to use mouse-operated dialog boxes to prompt for and accept input from the user. * Within a 4-D image, those edges that belong to a single hyperface in the original geometry can be highlighted. Highlighted edges are displayed in a distinct color which can be specified using the new COLOR_HILITE configuration parameter. HYPERGEO Version 2.1 Page F-3 * For 4-D images, all edges may be drawn using a color spectrum that maps segments of edge lines to a color that indicates their W-coordinate value in the original hyperobject. * A new dual imaging mode for 4-D to 3-D reductions displays both a projection and an intersection together side by side on the screen. When the dual mode is entered, HYPERGEO automatically rescales the display so the two images fit in the available area. All interactive commands affect both images simultaneously. The new DUAL parameter can be used in a HYPERGEO configuration file to select this imaging mode. * The user can change the position of the theoretical viewpoint used to calculate 4-D to 3-D projection images. This permits varying the amount of apparent perspective in the images from virtually none (equivalent to orthographic projection) to quite a lot. The 4-D projection viewpoint's position is controlled by the configuration parameter W_PROJECT plus several interactive commands, and is displayed in the menu area under the label W-PROJ. * A journalling capability permits capturing all graphics from a sequence of interactive manipulations of a geometrical model into a permanent file. A recorded journal file can be replayed in either forward or reverse order, and any particular frame of the journal may be viewed individually. * The current geometry can be positioned directly to any one of the set of distinct possible orthogonal orientations. These provide views that are perpendicular to one of the planes formed by a pair of coordinate axes. All twelve of the orthogonal orientations that are available for 4-D geometries can be generated, each via its own single-keystroke command. * A new "surface relief" display mode combines the features of hidden-line and stereoscopic display modes. A hidden-line display is shown using a red/blue anaglyph; with "3D" glasses the user views the front surface of the object with full depth perception of its relief. * A special state of the user interface can be entered which presents commands for temporarily adjusting the position in hyperspace of a selected vertex of the current hyperobject. The display is updated to show how moving the vertex in each of the four coordinate directions affects the resulting 3-D image. HYPERGEO Version 2.1 Page F-4 * A new "Locate" command permits specifying a precise orientation of the current geometry in terms of the displacement angles of its coordinate axes. * An alternative form of the Locate command is available in the vertex translation state of the user interface. It enables the direct entry of four coordinate values to position the selected vertex precisely at a particular point in hyperspace. * The HYPERGEO screen image can be written to a file in PCX graphics format. This permits using HYPERGEO images in documents produced by third-party software. * A DOS shell can be started from within HYPERGEO. This pauses the HYPERGEO program and permits running one or more DOS commands. When the DOS shell is exited, the HYPERGEO display is restored to its previous state. * The on-line help facility has been expanded from one page to eight pages. The new on-line help is organized hierarchically according to categories of related commands. * The number of separate pure shades used for solid surface rendering has been increased from eight to twelve; and... * ...color dithering techniques can be used to increase the effective number of separate shades available for solid surface rendering from twelve to twenty-three. The new configuration parameter SOLID_DITHER controls this option. * Command-line options have been added that permit overriding certain automatic hardware and system software status checks that are performed by HYPERGEO. These options cope with idiosyncratic CPUs and mouse drivers that misidentify themselves to the program. * The meaning of the scale factor has been mathematically inverted. It is now defined as screen inches per geometry unit (in Version 1 it was geometry units per screen inch). This makes the value more intuitively sensible; a larger scale factor now means a larger display, and vice versa. To prevent the misapplication of existing data, the configuration parameter that specifies the scale factor has been renamed; the old RATIO parameter no longer exists, and the new parameter is called SCALE. HYPERGEO Version 2.1 Page F-5 * The eye-to-screen parameter automatically restricts its value to a range that avoids numerical errors and corrupted graphics. New interactive commands permit setting the parameter directly to either its maximum or its minimum permissable value. * HYPERGEO can now use a Microsoft compatible mouse even if there is no standard mouse driver installed on the system. The program will detect and use a mouse on either serial port, COM1 or COM2. * For systems with VGA graphics, the default dark gray is redefined in the system palette to a shade that appears more reliably dark and gray. Note: The following two items are made available only with registered copies of HYPERGEO. They are not included in the shareware version. * Six utility programs have been created which automate the generation of geometry definition files for parameterized families of six types of curved object: cone, cylinder, sphere, hypercone, hypercylinder, and hypersphere. The programs calculate the geometry of shapes that are gridded approximations to these four curved shapes. The user can specify the fineness of the gridding. (See Appendix G for a fuller description.) * Additional geometry definition files have been created which describe the two "quasi-regular" 3-D convex polyhedra, the cuboctahedron and the icosidodecahedron, along with their derived compound and dual forms. HYPERGEO Version 2.1 Page G-1 Appendix G. Utility Programs for Geometry File Generation Six stand-alone utility programs have been developed which automate the creation of certain categories of geometry definition files. These utilities are supplied with registered versions of the HYPERGEO program. (They are not included in the shareware release.) These utilities all deal with objects that contain curved surfaces. They generate an approximation to the curves that consists of a gridded structure of triangular or rectangular polygons and polyhedra. The user specifies a parameter that controls the fineness of the gridding. There is an obvious trade-off: a finer gridding factor produces more accurate models of the curved surfaces, but it also produces larger geometry files that are slower to display and to manipulate. The six utilities generate gridded models of cones, hypercones, cylinders, hypercylinders, spheres, and hyperspheres. The three-dimensional members of this list - cones, cylinders and spheres - are the familiar objects of standard solid geometry. The hypercone, hypercylinder, and hypersphere are their four-dimensional analogs. A cone is defined as the shape formed by a circle and the surface that connects each point on the edge of the circle to a point not in the circle's plane. A cylinder is a shape formed from two circular disks that are parallel in 3-space and are connected by a surface that joins all corresponding points on their edges. A sphere is the surface of all points that are equidistant in 3-space from its center point. A hypercone consists of a sphere and the hypersurface that connects each point on the surface of the sphere to a point that is not on the sphere's hyperplane. A hypercylinder is formed by two spherical surfaces that are parallel in 4-space and are connected by a hypersurface that joins all of the sphere's corresponding points. A hypersphere is the set of all points that are equidistant in 4-space from its center point. For a sphere, the gridding used to approximate the curved surface corresponds exactly to the latitude and longitude lines used to define locations on the earth's surface. For the cone and cylinder, the base circles are approximated by regular polygons, and the connections to the vertexes of the polygons form the gridded approximation of the conical or cylindrical surface. For a hypercone or hypercylinder, the base spheres are themselves gridded spheres (with "latitude" and "longitude" lines). The connections of these gridded vertexes and edges form the hyperconical or hypercylindrical surface. In the hypersphere, the third gridding dimension spans a sequence of parallel spheres each of which is itself approximated with 3-D "latitude" and HYPERGEO Version 2.1 Page G-2 "longitude" measures. The connecting edges, faces, and hyperfaces between these spheres define the hyperspherical surface. The six utility programs are named as follows: CONE - Cones SPHERE - Spheres CYLINDER - Cylinders HYCONE - Hypercones HYSPHERE - Hyperspheres HYCYLNDR - Hypercylinders Each is executed with the same syntax, for example: SPHERE <gridding-parameter> The "gridding parameter" is an integer that must be at least 2. It determines the solid angle between parallel grid lines. This angle is equal to 180 degrees divided by the gridding parameter. Thus, a gridding parameter of 4 will generate grid lines that are 45 degrees apart on the object's surface. The larger the gridding parameter, the higher the accuracy of the model, and the larger and slower the file to execute. The output of the programs is a binary geometry definition that may be read directly into HYPERGEO. The name of the file is fixed for each program: CONE - CONEnn.BIN SPHERE - SPHEREnn.BIN CYLINDER - CYLNDRnn.BIN HYCONE - HYCONEnn.BIN HYSPHERE - HYSPHRnn.BIN HYCYLNDR - HYCYLRnn.BIN where 'nn' is the gridding parameter value (with leading '0' if less than 10). Thus, the command HYCYLNDR 3 produces a geometry definition file named HYCYLR03.BIN. This is a hypercylinder gridded at 60 degree intervals. The user should be forewarned that because of the inherent complexity of hyperspace, the geometry files for hypercones, hypercylinders, and hyperspheres get very large very quickly, even for relatively modest values of the gridding parameter. The hypersphere grows the most quickly; a gridding parameter of 4 is already large for a hypersphere, and you will only be able to handle a hypersphere with a gridding parameter of 5 if your system has a full 640K of memory, almost all of it available to HYPERGEO.