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help47.txt
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1995-10-31
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To start 'The Geometry Aide' version 1.0 type GEOMETRY at the DOS prompt
{make sure the program is in your path}. Once the program begins, accessing
the various functions is a matter of navigating the menus. Should you ever
find yourself in need of help while in the program select 'Help' (move the
green menu bar over 'Help' using the keyboard's arrow keys and hit ENTER)
from the main menu and then select 'Help' again on the following pull down
menu to bring up a selection of help topics. Select the area you wish to
know more about. Hit the ESC key at any time to cancel a menu and return
to the main menu. The following index is essentially what is included in the
help topics.
I. Menu Navigation
II. Symbols
III. 2-D Images
A. Triangle
B. Circle
C. Rectangle
D. Polygon
E. Parallelogram
F. Trapezoid
G. Ellipse
H. Parabola
I. Line
IV. 3-D Images
A. Pyramid
B. Sphere
C. Rectangular Solid
D. Ellipsoid
E. Cone
F. Cylinder
G. Line
V. Keyboard
VI. Plotters
VII. Desktop Options
VIII.Errors
I. MENU NAVIGATION:
Navigating through 'The Geometry Aide' is a simple process. By using the
right and left arrow keys on your keyboard you can move from subject to
subject on the main menu at the top of the screen. Use the up and down
arrow keys to move through the pull down menus. The enter key selects
whatever is currently highlighted on any menu. The escape key will exit
any submenu without making a selection. Two dots following any menu item
indicates that there is another menu or window following that selection.
To exit 'The Geometry Aide' select 'exit' under the file option of the
main menu.
II. SYMBOLS:
There are a number of symbols in mathematics that can be used to represent
the same operation; for example, <*> and <x> both represent multiplication.
There are other symbols that can be difficult to reproduce on the computer
such as the standard representation for exponents. The mathematical symbols
used in 'The Geometry Aide', where they differ from the norm or have more
than one representation, are listed below followed by an example.
'*' represents multiplication.
example: 2 * 3 = 6
'^' denotes that the following number is an exponent.
example: 4^2 = 16 'four squared is equal to sixteen.'
'/' represents division.
example: 4/2 = 2
'PI' represents a constant. This constant denotes the ratio of a circle's
circumference to its diameter.
example: PI can also be thought of as the number (3.14).
III. 2-D IMAGES:
A. Triangle:
A polygon of three sides is known as a triangle. There are many types of
triangles; see the below list for a definition of the different types.
Triangle Types:
Scalene: A scalene triangle is a triangle in which no two sides are equal.
Isosceles: An isosceles triangle is a triangle with at least two sides
equal.
Equilateral: An equilateral triangle is a triangle with all sides equal.
Acute: An acute triangle is a triangle with three acute angles (less
than 90 degrees).
Obtuse: An obtuse triangle is a triangle with an obtuse angle (greater
than 90 degrees).
Right: A right triangle is a triangle with a right angle (equal to 90
degrees).
Equiangular: An equiangular triangle is a triangle with all angles equal.
All triangles have several things in common: first; the sum of their
angles must equal 180 degrees; second, their area can be computed from
multiplying the base by the height and dividing the result by 2. In
'The Geometry Aide' the default triangle is an Isosceles triangle. Where
the sides of the triangle meet is known as the vertex.
When selecting 'Triangle' within the '2-D image' submenu a parameter
window will open up with the equation for the area of a triangle. Enter
the base and height of the triangle; the base is the length of the bottom
of the triangle and the height is the vertical distance from the bottom to
the upper vertex.
Try constructing other types of triangles other than an isosceles triangle
with the 2-d plotter. The default triangle can be moved with the arrow
keys and rotated around its center. See 'Keyboard' under the help menu
for an explanation of the key combinations.
B. Circle:
A circle is the set of points in a plane which are an equal distance from
a given point (also in the plane) called the center. The straight line
distance from the center to any of these points is known as the radius. The
straight line distance from a point on the circle's boundary, passing
through the center, to another point on the circle's boundary is known as
a diameter. The equation for the area of a circle is..
Area = PI * (radius)^2
The perimeter which can be thought of as the 'distance around' the circle
can be computed from..
Perimeter = 2 * PI * radius
The equation of a circle in rectangular coordinates is..
(X - X_center)^2 + (Y - Y_center)^2 = (radius)^2
X_center and Y_center represent the x and y coordinates of the center
respectively.
After selecting 'Circle' in the '2-D image' submenu a pop up window will
ask you to enter the radius of the circle. Enter your own or select the
the default value by hitting the 'Enter' key.
The circle will automatically graph itself with its center at (0, 0).
The circle can be moved around the coordinate axis with the arrow keys.
See 'Keyboard' under the help menu for an explanation of the key
combinations.
C. Rectangle:
A rectangle is a polygon of four sides. If all the sides are of equal
length the rectangle is called a square. The area of a rectangle is
found with the following equation..
Area = Length * Height
If the rectangle is a square the Length will be equal to the Height.
After selecting 'Rectangle' in the '2-D image' submenu a pop up window
will ask you to enter the Height and Length of the rectangle. Enter your
own or select the default values by hitting the 'Enter' key.
The rectangle will graph itself with its center at the origin of the axis.
The rectangle can be moved with the arrow keys and rotated about its axis.
See 'Keyboard' under the help menu for an explanation of the key
combinations.
D. Polygon:
A polygon is a figure in a plane which is bounded by straight lines. The
minimum number of sides for a polygon is three while there is no limit for
the maximum. A convex polygon has lines as sides which do not contain any
points within the interior of the polygon. A regular polygon is a convex
polygon with all sides and angles (created by the union of those sides)
equal. The default polygon for 'The Geometry Aide'is a regular polygon.
With all angles and sides of a regular polygon equal you could inscribe it
within a circle. You would find that the radius of that circle is also
the distance from the center of the polygon to the intersection of any two
sides.
When selecting 'Polygon' within the '2-D image' submenu a parameter window
will open up asking for the number of sides to give the polygon and the
'radius'. The term 'radius' refers to the radius a circle would have with
the polygon inscribed within it. Enter the number of sides and radius.
Try constructing other types of polygons, other than the default one
provided, with the 2-d plotter. The default polygon can be moved with
the arrow keys around its center. See 'Keyboard' under the help menu for
an explanation of the key combinations.
E. Parallelogram:
A parallelogram is a polygon of 4 sides in which pairs of opposite sides
are parallel. The area of a parallelogram can be found with the equation..
Area = side_1 * side_2 * sin (Θ)
'side_1' and 'side_2' represent adjacent sides and Θ represents the angle
formed by their intersection.
When selecting 'Parallelogram' within the '2-D image' submenu a parameter
window will pop open. Enter the length of the bottom and top sides along
with the height of the parallelogram. The height will determine the length
of the remaining sides. Enter the angle to be formed by the intersection
of the adjacent sides.
The default parallelogram can be moved with the arrow keys and rotated
around its center. See 'Keyboard' under the help menu for an explanation
of the key combinations.
F. Trapezoid:
A trapezoid is a polygon of 4 sides with exactly one pair of parallel
sides. The parallel sides are sometimes known as bases while the remaining
sides are known as legs. The area of a trapezoid can be found with the
equation..
Area = (1/2) * Height * (Base_1 + Base_2)
'Base_1' and 'Base_2' represent the base sides and 'Height' represents the
distance between the base sides.
When selecting 'Trapezoid' within the '2-D image' submenu a parameter
window will pop open. Enter the length of the bottom side along with the
height and the angles made by the sides with the bottom length. The height
will determine the length of the top side.
The trapezoid can be moved with the arrow keys and rotated around its
center. See 'Keyboard' under the help menu for an explanation of the key
combinations.
G. Ellipse:
An ellipse is a closed curve in the plane having two points within its
interior known as foci, such that the sum of the distances from these
foci, for any point on the boundary, is a constant. An ellipse can
sometimes look like a 'squashed' circle. The equation for an ellipse
in rectangular coordinates is..
( (X - X_c)^2 / (a)^2 ) + ( (Y - Y_c)^2 / (b)^2 ) = 1
'X_c' and 'Y_c' represent the x and y coordinates of the ellipse's center
respectively. 'a' and 'b' are the semi-axes for the ellipse. Whichever
one is larger is known as the semi-major axis while the other is the
semi-minor axis.
Selecting 'Ellipse' in the '2-D image' submenu will open a pop up window.
Enter values for the ellipse's semi-axes or accept the default values by
hitting the 'Enter' key. Notice the differences in the appearance of the
ellipse with differing values for its semi-axes.
The ellipse will start out with its center at the origin. The ellipse can
be moved around the coordinate axis with the arrow keys. See 'Keyboard'
under the help menu for an explanation of the key combinations.
H. Parabola:
A parabola is a curve such that any point on the curve is an equal distance
from a fixed point called the focus and a fixed line known as the
directrix. The vertex of a parabola refers to the point midway between the
directrix and focus. The equation of a parabola is..
(Y - Y_v) = (1 / 4 * k) * (X - X_v)^2
The x and y coordinates of the vertex is represented by X_v and Y_v. The
absolute value of k is the distance between the vertex and focus.
Selecting 'Parabola' in the '2-D image' submenu will open a pop up window.
Enter values for the coordinates of the parabola's vertex along with the
value for k.
The parabola can be moved around the coordinate axis with the arrow keys.
See 'Keyboard' under the help menu for an explanation of the key
combinations.
I. Line:
A line consists of at least two points; it extends indefinitely into the
plane. Each line has a slope which is calculated by dividing the 'rise'
with the 'run'. If (X1, Y1) and (X2, Y2) are two points on the line the
slope can be found with the following..
slope = (Y2 - Y1) / (X2 - X1)
A horizontal line has a slope equal to zero while a vertical line has no
slope at all. A line that falls moving from the left to the right will
have a negative slope; if the line rises moving from the left to the right
the slope will be positive. A line can be represented by several different
equations. First, is the slope-intercept form given by..
y = slope * x + b
The value b is the y intercept of the line. The general equation of a line
is..
Ax + By + C = 0
A, B and C are constants where A and B are not both equal to zero. The
point-slope form of a line containing point (x1, y1) is..
y - y1 = slope * (x - x1)
Which equation you use depends upon the information you have. 'The
Geometry Aide' uses the slope intercept form when describing a line.
After selecting 'Line' in the '2-D image' submenu a pop up window will
ask you to enter two points lying on the line. Enter your own points or
select the default values by hitting the 'Enter' key.
The line can be moved around the coordinate axis with the arrow keys.
See 'Keyboard' under the help menu for an explanation of the key
combinations.
IV. 3-D IMAGES:
A. Pyramid:
A pyramid is a solid having a polygon as its base and for its sides
triangles with a common vertex. The triangular sides of a pyramid are
known as lateral faces. The vertical distance from the common vertex to
the base is known as the height. The default pyramid for 'The Geometry
Aide' is a regular pyramid. Regular pyramids have a regular polygon (all
sides of the polygon are equal) for a base and its faces are made up of
isosceles triangles. In 'The Geometry Aide' the default pyramid has a
square for its base.
After selecting 'Pyramid' in the '3-D image' submenu a pop up window will
ask you to enter the height and base length of the pyramid. Enter your
own or select the the default values by hitting the 'Enter' key.
If you wish to create your own pyramid try plotting it on the 3-D plotter
under the 'Plotters' option of the main menu. The default pyramid can be
moved with the arrow keys and rotated around any axis. See 'Keyboard'
under the help menu for an explanation of the key combinations.
B. Sphere:
A sphere is the set of points which are an equal distance from a given
point called the center. The straight line distance from the center to
any of these points is known as the radius. The straight line distance
from a point on the sphere's boundary, passing through the center, to
another point on the sphere's boundary is known as a diameter. Notice
how closely the definition of a sphere is to that of a circle. While the
definition of a circle qualifies the set of points an equal distance from
the center as lying in a plane, the definition of a sphere does not. The
equation for the volume of a sphere is...
Volume = (4/3) * PI * (radius)^3
The equation of a sphere in rectangular coordinates is..
(X - X_c)^2 + (Y - Y_c)^2 + (Z - Z_c)^2 = (radius)^2
X_c, Y_c and Z_c represent the x, y and z coordinates of the center
respectively.
After selecting 'Sphere' in the '3-D image' submenu a pop up window will
ask you to enter the radius and center coordinates of the sphere. Enter
your own or select the default values by hitting the 'Enter' key.
The sphere will graph itself with its center at the entered coordinates.
The sphere can be moved around the coordinate axis with the arrow keys.
See 'Keyboard' under the help menu for an explanation of the key
combinations.
C. Rectangular Solid:
A rectangular solid has six faces. Each face is bounded by a rectangle.
An edge is formed with the intersection of two faces. If all the edges
are of equal length then the rectangular solid is known as a cube. The
volume of a rectangular solid can be found with the equation..
Volume = Length * Width * Height
After selecting 'Rectangular solid' in the '3-D image' submenu a pop up
window will ask you to enter the length, width and height of the
rectangular solid in order to alter its volume. Enter new values or
select the defaults by hitting the 'Enter' key.
The rectangular solid will appear with its center at the origin of the
axis. The rectangular solid can be moved with the arrow keys and rotated
about any axis. See 'Keyboard' under the help menu for an explanation of
the key combinations.
D. Ellipsoid:
An ellipsoid is a surface in three dimensional space such that any plane
section forms an ellipse. The equation of an ellipsoid in rectangular
coordinates is..
((X-X_c)^2/a^2) + ((Y-Y_c)^2/b^2) + ((Z-Z_c)^2/c^2) = 1
X_c, Y_c and Z_c represent the x, y, and z coordinates of the ellipsoid's
center respectfully. An ellipsoid, like the ellipse, has semi-axes; a, b,
and c each represent a semi-axis along the x, y and z axis.
After selecting 'Ellipsoid' in the '3-D image' submenu a pop up window
will open. Enter the values of the ellipsoid's center coordinates along
with the semi-axes. You may accept the default values by hitting the
'Enter' key.
The ellipsoid can be moved with the arrow keys. See 'Keyboard' under the
help menu for an explanation of the key combinations.
E. Cone:
A cone consists of a circular base and a surface composed of line segments
joining every point on the base boundary to a common vertex. 'The Geometry
Aide' deals only with right circular cones. A right circular cone has a
vertical height which stretches from its base to its vertex. Also, all
line segments drawn from the vertex to the base boundary are equal. The
volume of a right circular cone is ..
volume = 1/3 * PI * (radius)^2 * height
'radius' refers to the radius of the circular base and 'height' stands for
the vertical distance from the base to the vertex.
The default cone for 'The Geometry Aide' is a right circular cone.
Selecting 'Right circular cone' under the '3-D image' submenu will bring
up a pop up window. Enter the base radius and the height along with the
starting coordinates for the cone's center point (this is the point located
in the center of the base and halfway between the vertex and base).
The default cone can be moved with the arrow keys. See 'Keyboard' under
the help menu for an explanation of the key commands.
F. Cylinder:
A right circular cylinder is the set of all lines perpindicular to the
plane of the circle passing through the circle's boundary. The volume of a
cylinder is given by..
Volume = PI * (radius)^2 * height
The term height refers to the distance from the cylinder's base to its
top.
Selecting 'Right circular cylinder' from the '3-D image' submenu will
bring up a parameter window. The term 'center' in the 'The Geometry Aide'
as it pertains to a right circular cylinder refers to the point at the
center of the circular base and halfway between the base and the top.
Enter the x, y, and z coordinates of the cylinder's center point along with
the radius.
The cylinder will graph itself with its center at the entered coordinates.
The cylinder can be moved around the coordinate axis with the arrow keys.
See 'Keyboard' under the help menu for an explanation of the key
combinations.
G. Line:
A line in three dimensional space has the same characteristics as a line
in a plane. The only difference being that the 3-D line extends
indefinitely into space as opposed to a plane. Be sure you understand the
concepts of slope for a line in a plane before taking on a three
dimensional line. 'The Geometry Aide' uses simultaneous equations (if one
holds true the others must be true too) in describing a line in space.
These equations are..
(X - X1) / (X2 - X1) = (Y - Y1) / (Y2 - Y1)
(Y - Y1) / (Y2 - Y1) = (Z - Z1) / (Z2 - Z1)
X1, Y1, Z1 and X2, Y2, Z2 are coordinates for two separate points on the
line. Simultaneous equations are good only if Y2 is not equal to Y1, X2
is not equal to X1 and Z2 is not equal to Z1. In 3-d space the equation
for a line is best represented by parametric equations which are beyond
the scope of 'The Geometry Aide'.
After selecting '3d-line' in the '3-D image' submenu a pop up window will
ask you to enter two points lying on the line. Enter your own points or
select the default values by hitting the 'Enter' key.
The line cannot be moved around the coordinate axis with the arrow keys.
V. KEYBOARD:
Some of the images in 'The Geometry Aide' can be moved or rotated using key
combinations from the keyboard. Each image can have its own set of unique
key combinations so it is important to review these combinations before
attempting to graph an image. Listed below are the images and the key
combinations which can be used when manipulating them. If you plan on
using the arrow keys on your keypad you should make sure your <NUM_LOCK>
key is off. When referring to the keyboard keys the following nomenclature
will be used..
ENTER = Enter key.
TAB = Tab key.
SHIFT = Shift key.
RT_ARROW = Right arrow key.
LF_ARROW = Left arrow key.
UP_ARROW = Up arrow key.
DN_ARROW = Down arrow key.
CTRL = Control key.
ESC = Escape key.
'key' = Key on keyboard.
Navigating Pop Up Windows:
The pop up windows that follow the selection of any image under
'2-D image' or '3-D image' usually contain several fields pertaining to
the image. If the field values are not changed the default values will
be used. Pressing the ENTER key accepts the values within the chosen
field; if there are other fields left the cursor will automatically move
to the next one to await input. Pressing TAB will also move the cursor to
the next available field while SHIFT + TAB will move the cursor to the
previous field.
Exiting:
After entering all the parameters in the pop up window the image will be
graphed. You may then interact with the image by moving or rotating it.
To quit image interaction and return to the main menu press the ESC key.
Image Movement:
The following list breaks down the key combinations used for the different
images.
Valid key combinations for..
Triangle
Rectangle
Parallelogram
Trapezoid
----------------------------------------------------
RT_ARROW = Move image to the right.
LF_ARROW = Move image to the left.
UP_ARROW = Move image up.
DN_ARROW = Move image down.
CTRL + RT_ARROW = Rotate image clockwise about axis.
CTRL + LF_ARROW = Rotate image counterclockwise about axis.
'r' = Reset image.
ESC = Return to main menu.
Valid key combinations for..
Circle
Polygon
Ellipse
Parabola
Line
----------------------------------------------------
RT_ARROW = Move image to the right.
LF_ARROW = Move image to the left.
UP_ARROW = Move image up.
DN_ARROW = Move image down.
'r' = Reset image.
ESC = Return to main menu.
Valid key combinations for..
Pyramid
Rectangular solid
----------------------------------------------------
RT_ARROW = Move image to the right.
LF_ARROW = Move image to the left.
UP_ARROW = Move image up.
DN_ARROW = Move image down.
'.' = Move image away (decreasing z).
',' = Move image closer (increasing z).
CTRL + RT_ARROW = Rotate image about y axis.
CTRL + LF_ARROW = Rotate image about y axis.
CTRL + UP_ARROW = Rotate image about x axis.
CTRL + DN_ARROW = Rotate image about x axis.
CTRL + '.' = Rotate image about z axis.
CTRL + ',' = Rotate image about z axis.
'r' = Reset image.
ESC = Return to main menu.
Valid key combinations for..
Sphere
Right circular cylinder
Right circular cone
Ellipsoid
-----------------------------------------------------
**Note: Due to the amount of processing time it takes to graph an image
you must first 'erase' it before moving it around the
coordinate axis. Hit 'e' to erase the image; it will disappear
and a pixel will take its place. Move the pixel to where you
wish to regraph the image and strike 'd' to display the image
once again. The image should be displayed before hitting the
ESC key to exit back to the main menu.
'e' = Erase image.
'd' = Display image after erasing.
RT_ARROW = Move pixel to the right.
LF_ARROW = Move pixel to the left.
UP_ARROW = Move pixel up.
DN_ARROW = Move pixel down.
'.' = Move image away (decreasing z).
',' = Move image closer (increasing z).
'r' = Reset image.
ESC = Return to main menu. Note: if the image is erased you will have
to hit the escape key twice to return to the main menu.
Valid key combinations for..
3d-line
-----------------------------------------------------
There are no key combinations as the 3d-line cannot be moved.
VI. PLOTTERS:
The '2-d plotter' and '3-d plotter' options under the 'Plotters' selection
on the main menu allows for the creation of custom two and three
dimensional wire frame images. Once one of these options is selected a
'point plotter' will appear in the upper left hand corner of the desktop.
The 'point plotter' takes the first point entered and plots a point at
that location; after the first point is plotted lines are drawn between
the following entered points and the previous point plotted. A maximum
of 50 points may be plotted. You may quit plotting at any time by
toggling the 'on' or 'off' option by hitting 'p' on the keyboard while the
'point plotter' is displayed.
VII.DESKTOP OPTIONS:
The desktop colors in 'The Geometry Aide' can be changed by choosing
'Options' under the main menu and then selecting 'Colors' on the following
submenu. Within the colors menu are selections for 'Axis', 'Windows',
'Windows text', 'Image foreground', and 'Equations'. By selecting one of
these options you will be able to control the desktop colors. The
'Restore defaults' option restores the program's original color settings.
Once you have set the desktop colors to your preference you can have 'The
Geometry Aide' use these choices every time it is started up by selecting
'Save settings' in the submenu under the 'File' option of the main menu.
The 'Repaint desktop' option redraws the axis; while moving the images
around the desktop portions of the axis can become pocked with gaps,
repainting eliminates this effect.
Displaying the axis can also be controlled from the main menu. A two or
three dimensional axis is drawn for each image graphed. If you wish to
turn the axis 'off' select 'Options' under the main menu and then choose
'Axis' under the subsequent pull down menu. A pop up window in the middle
of the screen will ask if you wish to toggle the axis 'on' or 'off'.
Strike 'x' on the keyboard to switch from 'on' to 'off'. By selecting
the 'off' option all images will be graphed without displaying their
associated axis.
Some of the images in 'The Geometry Aide' can be rotated around their
axes by using an arrow key combination. The number of degrees to be
rotated can be set by selecting 'Options' under the main menu and then
choosing 'Rotation angle' on the subsequent pull down menu. The default
value is 10 degrees; this means that every time a particular key
combination is entered the object is rotated 10 degrees around the
appropriate axis. When selecting rotation angles try avoiding values
which are multiples of 360 ( a full rotation ) as the image will rotate
but there will be no apparent visual effect. See 'Keyboard' under the
help option for an explanation of arrow key combinations.
VIII.ERRORS:
There are basically three types of error messages that can be generated
in 'The Geometry Aide'. The first two concern movement and the entry
of parameters for the images. The third deals with hardware errors
generated outside of the program on the local system. Moving an image off
the screen will usually generate an error message and reset the image. On
a three dimensional axis it is possible to move the image off of the screen
without generating this error message. For example, on a two dimensional
axis the value x=16 is not visible regardless of the value of y. A three
dimensional axis can have x=-20, y=0, and z=-20 and still be visible,
therefore the error checking on a three dimensional axis is checking for
the most extreme values only.
Entering invalid or out of range parameters for images will also generate
an error message; you will be prompted to reenter these values. An invalid
value can be generated by entering a zero value in a field which represents
the denominator in a fraction. An out of range value is one that would
make graphing the image an impossibility due to its size or range.
Hardware errors are generated due to a fault within the local system. This
could be anything as extreme as failure of internal hardware or as minor as
the removal of a floppy disk from its drive.