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************************************************* * * * * ************* QMJ version 5.31a *************** * * * * ************************************************* A Shareware program to display the Mandelbrot Set, Complex and Quaternion Julia sets on the Amiga. Written by Chris Baxter in Hong Kong and Naples 1989-91. All Rights Reserved. Version 5.31a. Written in C and compiled with the Lattice C compiler version 5.05. The program may only be distributed with this notice and may not be published or sold without the author's prior consent. Neither the program nor the documentation may be altered in any way during its distribution or use. No fee may be taken for this program other than a minimal copying and media charge. The images on the distribution disk were produced on a standard PAL A500 with 1 megabyte of memory. The full version of this program requires 1 megabyte to run; a cutdown version for 512k machines is available from the author, as is a large number of fractal images. ************************************************************************* Based on ideas in these articles in 'Scientific American':- Aug. 1985 "Computer Recreations" p8 by A.K.Dewdney Nov. 1987 "Computer Recreations" p118 by A.K.Dewdney Feb. 1989 "Computer Recreations" p88 by A.K.Dewdney and this one in 'Fractal Report':- Issue 3 1989 "Quaternion Julia Sets" p2 by Dr Ian Entwhistle The Mathematical Theory ======================= The Mandelbrot Set and Julia Sets are drawn by their coordinates in the Gaussian plane. The sets themselves appear in classical black in this program; it is the boundaries approaching the sets which are coloured and which provide the eye-catching image. Membership of the sets is a function of two variables: F(z,c): z -> z^2 + c where z and c are both complex numbers consisting of both a real and an imaginary part; the function is plotted by these two components, the imaginary axis being vertical (y) and the real one horizontal (x). The same function is used for both the Mandelbrot & Julia sets. The Mandelbrot Set is generated by setting z to zero initially and varying the value of c, therefore the points on the plane relate to the real & imaginary components of c. The Julia sets arise from fixing c and varying the value of z, so that the plotted points relate to the real and imaginary parts of the value of z. The Quaternion Julia set images are 2D slices through the 4D sets. The function that describes them is: F(Q,q): Q -> Q^2 +q Each of Q and q consist of a real part and components on the i, j and k axes of the 4D space. According to Hamiltonian algebra Q may be squared as follows: since Q = a+bi+cj+dk therefore Q^2 = (a+bi+cj+dk)(a+bi+cj+dk) = a^2+abi+acj+adk+bia+(bi)^2+bicj+bidk+cja+(cj)^2+ cjdk+dka+dkbi+dkcj+(dk)^2 but ij=ik=jk=ki=kj=0 (scalar multiplication) therefore Q^2 = a^2+2abi+2acj+2adk+(bi)^2+(cj)^2+(dk)^2 but i^2 = j^2=k^2=-1 therefore Q^2 = a^2-b^2-c^2-d^2+2abi+2acj+2adk Separating Q^2+q into real and imaginary components gives: 1. Qreal = a^2-b^2-c^2-d^2+qa 2. Qimag.i=2ab+qb 3. Qimag.j=2ac+qc 4. Qimag.k=2ad+qd For a 2D image the program can only use two planes to provide the screen mapping; in this program those corresponding to equations 1 and 3 are used. It has been shown that if the size of either function ever reaches 2 then it will continue towards infinity and the initial values used (depending on the type of set) do not lie within one of the sets. The colours of the images reflect the degree of boundedness that point has. To determine membership of the sets thousands of iterations might be required; the maximum iteration in this program (which means membership of the sets, if the size is still less than 2) may be varied to suit real time constraints - Dewdney reports that 100 may often be adequate. However, in practice it may be useful to preview an image using a low maximum iteration and small image size; if all is well reset the parameters and generate a larger image. Of course, closer to the sets the number of iterations required increases, so that images of these regions are slower to produce. On first loading an ffp produced image will be flagged as such to the user; an 'ieee' flag in the title bar tells you that the image was created in double precision maths. (Such flagging subsequently changes to suit the program actually being run.) Please Note ----------- The fast floating point maths option of the program seems accurate enough when working to about 6 or 7 significant figures; beyond these limits use the standard double precision maths. Generating these images is heavy on floating point maths, so many of them can take some time to finish. With version 5.31 of this program you can:- ** generate images of the Mandelbrot Set ** generate images of the Julia Sets ** generate images of Quaternion Julia slices ** specify with the mouse which part of an image to enlarge ** save an image (compressed or uncompressed ILBM) ** reload an image (compressed/uncompressed ILBM format) ** continue a saved, but unfinished, image ** edit the colours used for drawing ** load only the palette from another image ** control the complete mathematical process ** choose the method for colouring the image ** send the image to the printer ** choose IEEE or FFP maths ** choose low or medium resolution modes History ------- In version 5.0 several bugs were fixed and improvements made over v4.0 and earlier versions:- QMJ project icon startup was included saving now also saved an accompanying image icon the framing procedure was improved palette editing was enhanced the parameter screen was speeded up QMJ now ran as a background process linked with cback.o 5.10 Provided a choice of IEEE or FFP maths. The former are high precision routines that will use a maths coprocessor if it is present; the latter use the Motorola Fast Floating Point routines which are faster, but less accurate. An IEEE double is held in 64 bits; an FFP in only 32, so it clearly holds less information. The colour editing option was made to work during PAUSE. Loading a palette didn't blank the screen anymore. 5.20 used the MicroSmiths' file requester and returned colour editing to a draggable window, which is where it belongs. 5.30 prototyping & registerised parameters to speed things up. A new LowRes quick drawing mode. 5.31 Intro screen added. Help beefed up. Parameters speeded up. Fuller keyboard command support added. 5.31a uses a new file requester. Workbench 2 compatible. ** All versions PAL/NTSC compatible. When the program is run it detects the host video format and adapts itself to PAL or NTSC screen dimensions. The images supplied on the distribution disk are PAL and may require a full PAL screen (except the NTSC default image). If you've got an NTSC Amiga they either won't fit or they'll be distorted. Sorry about that. *************************************************************************** v5.31a Copyright (c) Chris Baxter October 1992 20 The Ridgeway Droitwich Worcs. WR9 8QB England The program & notes updated in Bilbao, Spain September 1992. You can contact me there until July 1994 at this address: calle Trauco 1, 3° izq 48007 Bilbao España ************************************************************************** Disclaimer ========== This program is shareware and it may not be sold or distributed without its copyright notices. However, you use the code entirely at your own risk. No warranties are expressed or implied as to its fitness for any purpose. If you like the program and keep it, please send me US $10 (or equivalent), this will make you a registered user to whom the source code can be made available. Alternatively, send me some of your fractal work - programs and/or images. The program is in C and was compiled with Lattice C version 5.05. It's taken me a long time to get it into shape, so your support would be appreciated. Please report any bugs or suggestions so that they can be incorporated in future releases. Please feel free to contact me on any Amiga programming topic. I would, of course, be happy to receive any interesting images generated with this program. *************************************************************************** Please Note: For the program to run you must have mathieeedoubbas.library mathieeedountrans.library in the libs: directory. The distribution disk also uses MuchMorePoPa by Fridtjof Siebert of AMOK and BootIntro by Roger Fischlin - the author is grateful to them for their PD work. *************************************************************************** How to start the program: ======================== either From WorkBench click on the appropriate QMJ icon or From the CLI type either qmj_pal or qmj_ntsc and the program will start. From WorkBench, if you have a previously saved QMJ image, click on the image icon and QMJ will automatically start up and then load the image. How to use the program: ====================== The program starts up with an introduction to itself and the control keys it uses; hit any key (except HELP) or a mouse button to move on. Then the Mandelbrot Set (PAL or NTSC, depending on your system) will be displayed, unless you started the program from a project icon. The program is now waiting further instructions from you. You can either load or draw an image at this stage or change some of the program's parameters. The PROJECT menu ================ SAVE will save the current screen image (subject to its current dimensions) and all of the parameters for generating the image. The image may be a standard IFF image or compressed according to the cmpByteRun1 run compression algorithm. (see Parameters) The parameters are held in a special chunk named 'QMJD', this is not a recognised IFF chunk. Other IFF readers should simply skip this chunk, if they are well written, so images should load into art programs etc. An icon will also be saved for the image; this is a project icon and will allow the image to be loaded automatically by double-clicking on it. ** See File Requester below. LOAD will ask for an ILBM (compressed or uncompressed) filename. If the ILBM image was not generated directly by QJM, an error will be reported, although the image may still load. This enables image 'processing' in DeluxePaint, for example, although you will need to re-type the image parameters (if you have a note of them) for them to be correct; this is because art programs and so on don't save the special parameter chunk. From version 5.0 an image can be loaded by clicking on its icon. If a version 4.0 (or earlier) image has no icon, copy an icon .info file generated under version 5.0 (or later) or the icon from some other project file (but don't forget to change the default tool). If the autoload of a project image fails, check the DEFAULT TOOL associated with the icon. ** See File Requester below. CLEAR will clear the screen. ABOUT gives a copyright notice. The program is SHAREWARE, not public domain. The author retains full copyright on source and executable code. The executable program must not be distributed without the full document- ation and copyright notices. PRINT will provide a screen dump of the current image to the printer. It will also print out the image's parameters. QUIT takes you out of QMJ via a requester to check you haven't goofed. The DRAW menu ============= DRAW starts drawing an image. When the program is quiescent, waiting for a command, this will be the only option available on this menu. When an image is being produced, the DRAW option will be disabled and PAUSE and STOP will be possibilities. STOP halts the drawing of an image. If you SAVE an image that has been STOPped before completion and then reload it, the DRAW command will continue the image at the start of the current line. There may be a slight delay, if you have stopped and restarted an image in mid-line, while the program redraws the start of that line again. This option is useful for producing lengthy images, when you might not wish to leave the computer on; simply save the image and continue it later. PAUSE will halt the program, disabling all menu choices other than COLOUR (on the CHANGE menu) and DRAW. When the program's drawing is paused, you may change the palette using COLOUR on the CHANGE menu. To restart drawing select DRAW again; QMJ will pick up on the current image where it left off. The CHANGE menu =============== PARAMETERS calls up the program's control screen. At the top the current function to be iterated is indicated. Beneath it are the current parameters and their values. To alter one of these, click in the text area and simply type in or edit the values. The parameters available are: For Mandelbrots - the real & imaginary parts of c. For Complex Julias - the real and imaginary parts of z and c. For Quaternion Slices - the values a,b,c,d and qa,qb,qc,qd where Q=a+bi+cj+dk and q=qa+qbi+qcj+qdk; a and qa are real, all other parts are imaginary; i,j,k are vectors representing the square root of -1. Only the parameters required for the chosen function are available for input and editing, the others are ghosted out. To move to the next parameter, simply hit return or click with the mouse on the chosen gadget. Beneath these mathematical parameters are the current screen dimensions (in pixels). The maximum screen size is 640x246 on a PAL machine or 640x190 on an NTSC screen. (The other 10 pixels are used for the title area.) At the bottom, on the left, there is the number of iterations to be performed before membership of the Mandelbrot or Julia sets is assumed. Any point that is iterated this number of times without its size reaching 2 will be coloured black on the screen. On the right are 4 gadgets. At the top there is the gadget to specify the mode for colouring pixels. The possibilities are 'modulo', 'nonlinear' and 'linear'. Modulo shading means that when the function size reaches 2 for the point being iterated (ie. it doesn't reach the maximum number of iterations) the colour is determined by the number of iterations MOD 14 (fourteen being the number of colours available). Each colour can represent several numbers of iterations under this scheme and it is often possible to see the range of colours repeated within the image. Linear implies an even, arithmetic spread of the 14 colours over the range up to the maximum number of iterations allowed. Nonlinear will spread the colours in a geometric progression over the given range of iterations, so that colours are more finely allocated further from membership of the sets. The second gadget on the right selects the function to be iterated, MANDELBROT, complex JULIA or QUATERNION Julia slice. The third gadget determines whether images will be saved compressed (in cmpByteRun1 run encoding) or uncompressed. The final gadget RETURNs you to the main screen. COLOUR will call the palette to allow editing of the colours the program uses for drawing. There are 14 colours used for drawing (plus black, used to show membership of a set in the 'traditional' way) and any of them may be altered as desired using Proportional Gadgets for their RGB components. Simply drag the RGB sliders to the intensity that you desire whilst watching the sample box displaying the current colour. The 16th colour is white and is used for titles etc. When you call up the parameters window, for example, QMJ will use false colours to make sure that you can read it. FRAME allows the user to select part of an image to investigate in greater detail. When you select FRAME you will be asked to click on the centre of the upper boundary of the new area. When you have done this you'll be asked to click on the centre of the lower boundary. The actual rectangle that you select will use these upper and lower bounds and the centre line you have indicated to produce an image appropriate to the current image proportions indicated on the parameter screen. (Note that the centre position isthe average of the upper and lower x positions when you clicked the left mouse button.) It's quite easy as the selection box will follow the mouse around, given the above constraints! At present there is no graphical way to select an image larger than the current one, other than going to the parameter screen and manually altering the parameters. SHOW TITLE will toggle between displaying the title-bar and hiding it. This can be useful when photographing the screen, for example. RESTORE DATA restores the data (available in the 'parameters window') to its state at the last LOAD or DRAW. This means that if you have an image loaded and make a mistake in framing, for example, you can get back to where you were before. CLEAR leaves you with only your current data and no old data to go back to. Generally this means that you can usually get back the parameters to suit the image on the screen. LOAD PALETTE will load the colour settings from another image without loading the image itself. FFFP / IEEE causes the program to change its maths precision. The default is IEEE so the menu will offer you FFP; if you select this item the program will draw faster, but less accurately. The menu wil show IEEE for the next selection; choose it again to return to double precision maths. LOWRES/MEDRES selects the resolution of the drawing. LowRes is almost 4 times as fast as MedRes, as it plots 4 points per iteration ie. double points in both x & y directions. It is useful to get a quick idea of how a plot will look. RESET ALL returns the program to its initial startup state, possibly including a default screen, depending on the version. ** File Requester Version 5.31a uses a different file requester from previous versions of QMJ. The display at LOAD or SAVE will consist of two windows. On the left there are the normal gadgets to LOAD or SAVE, CANCEL or type in the path and file names required. On the right there is a separate file list window showing ONLY the directories and files with a ".qmj" extension in the current directory. To open a directory click ONCE on the directory name and it will be added to the path name on the left and its contents displayed. To go back a level click on "<<PARENT" at the top of the screen. Files can be selected in a similar way. The requester will not lose its contents between LOADs and SAVEs (to save you time). It will detect a change of disk in the current drive if the requester is up. If you need to refresh the display at any other time select "GET DIR". (N.B. RAMB0: is the RAD: unit name under normal conditions. If you want to change these use a binary file editor. What? You don't have one? Check out the PD libraries or contact me for a copy of AFEd - The Amiga File Editor.) ALL COMMANDS have alternate keyboard sequences: ============================================== SAVE Left-Amiga & S or just F5 LOAD Left-Amiga & L or just F6 CLEAR Left-Amiga & X or just F2 ABOUT Left-Amiga & A or just HELP PRINT Left-Amiga & W QUIT Left-Amiga & Q or just ESC DRAW Left-Amiga & D STOP Left-Amiga & . PAUSE Left-Amiga & P PARAMETERS Left-Amiga & Z or just F3 COLOUR Left-Amiga & C or just F4 FRAME Left-Amiga & F or just F1 SHOW TITLE Left-Amiga & T or just F10 RESTORE DATA Left-Amiga & U LOAD PALETTE Left-Amiga & K or just F7 FFP / IEEE Left-Amiga & M or just F8 LOWRES / MEDRES Left-Amiga & O or just F9 RESET ALL Left-Amiga & R Don't forget that AutoRequesters can be answered positively with Left-Amiga V and negatively with Right-Amiga B. I hope you enjoy using the program. Chris Baxter October 1992, Bilbao