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 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 (é) 'side_1' and 'side_2' represent adjacent sides and é 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 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.