About: NMG93.SDK and OMG86.SDK
Distribution by: Charles T. turley

From Author:  Dr. Hank Levinson
Surface Mail: P.O. Box 83
              Paramus, N.J. 07653
                
Email: hlevinsn@andromeda.rutgers.edu 

Status: Copyrighted - freeware 
(may not be distributed in any manner for profit - must be distributed only for FREE)

NMG93.SDK and OMG86.SDK are both Shrinkit DOS 3.3 archives, which will unshrink to a
5.25 Apple II - DOS 3.3 disk.

The Documentation files for the above programs follow.

            DOCUMENTATION FOR THE NEW MATHEMATICAL GRAPHICS PROGRAMS

A - GRAPH.Y=F(X): This is almost exactly like the former program except for two
    additional features. Discontinuities will automatically not be graphed and
    will not disrrupt computation, and the user will be offered the choice of
    specifying a coordinate grid.

B - POLAR.PLOT.R=F(A): This polar plotting program plots R = F(A). After
    entering F(A) and the domain interval for A, the user is asked how many
    points should be plotted. Usually 360 will do; if there are rapid changes
    in R, more points may be specified. Discontinuities in F(A) will be ignored
    (the program will not plot and simply skip over them) and it will
    automatically calculate the range of F(A). The user may select to plot a
    polar coordinate grid after the function is plotted.

C - POLAR.PLOT.R=F(A).ARB.R: Here the user may select minimum and maximum
    values of the radius R. In B, everything was plotted from the origin to the
    maximum value of R. This program plots whatever sector of an annulus is
    specified. Again there is provision for automatically skipping over
    discontinuities, and for plotting coordinate grids.

D - PARAMETRIC.PLOT.X(T).Y(T): Automatic range calculation, discontinuity
    skipping and coordinate-grid plotting are additional features of this
    version of the previous parametric plotting program.

E - GRAPH.Z=F(X&Y): This program plots a function of 2 independant variables, X
    and Y, in 3-D. It uses only parallel perspective and has no provision for
    hidden point removal. It does have an automatic range feature, but that
    runs very slowly. There is no skipping over discontinuities; the program
    simply crashes at a discontinuity.

F - GRAPH.Z=F(X&Y).HDN.PNT: This program is like the previous one, but it does
    have a hidden point removal option which works as follows. A line is drawn
    from a point to be plotted orthogonally to the viewer. If this line passes
    through the surface being plotted, then the point being tested is "hidden"
    and is not plotted. To test whether the line from the point passes through
    the surface, the program computes values of Z both on the line and on the
    surface. These values are spaced along the viewing line. If the spacing is
    too wide, a passage through the surface may remain undetected. However the
    closer the spacing, the slower the program runs.
    A good example to see the use of this program is to graph the equation

                                    X*X - Y*Y
                        Z = X * Y * --------- .
                                    X*X + Y*Y

    Enter X*Y*(X*X-Y*Y)/(X*X+Y*Y+.0001) for F(X,Y). The .0001 is to avoid a 0
    denominator. Let X and Y each be confined to [-2,2] intervals, and Z to
    [-1.25,1.25]. Choose -30 for VA and 25 for VB. Select the hidden point
    option, use a "T-interval" of .2 to search along lines from points towards
    the viewing plane, and 15 intervals in X and Y directions with 250 points
    per line. The program plots very slowly. To see the surface from "the other
    side", enter -.2 for the T-interval. 

G - GRAPHICAL.ANALYSIS: This is a classical tool to study the dynamics of
    functions of a single real variable, finding attracting points and
    repelling fixed points with respect to iterating the function. The user is
    allowed to enter any function and a domain and range. The program will plot
    the function and the line Y = X. Then the user will be prompted for the
    X-coordinate of a point to be tested. The result is a diagram showing the
    point lifted vertically to the curve, then horizontaly to Y = X, then
    vertically to the curve, etc. until the point either escapes from the
    domain-range rectangle, or is trapped by some attracting point.
    For a clear elementary treatment of this and the rest of the dynamical
    analysis programs, the reader is referred to "Chaos, Fractals and Dynamics"
    by Robert L. Devaney, Addison-Wesley Pub. (1990).

H - ORBIT.DIAGRAM.VAR.DOMAIN: Here the dynamical properties of the function
    X*X + C, where C is a constant, is analysed for various values of C. A
    domain for C is chosen (somewhere between -2 and 0.25) and the orbit
    diagram is graphed. It is possible to carefully select domains which show
    various periodicity windows in which attracting orbits bifurcate. One may
    also see a fundamental fractal behavior in which, under magnification, the
    same structures appear (although somewhat distorted) in the small as in
    the large. The user is again referred to Devaney's book.

I - JULIA.ARB.FCT: The Julia set of a function is the boundary between the set
    of points which escape under iteration of the function, and those which
    have finite orbits under the iteration. (It is named after the French
    mathematician Gaston Julia, who lived at the end of the 19th century,
    defined the concept, but never saw the realization of his idea graphically;
    the technology simply was not available then.) The functions considered are
    of a complex variable Z = X + iY. The real and imaginary parts of the
    function to be explored are entered, together with the center point of a
    square region in the complex plane. A magnification factor is also entered,
    and the program iterates the function at points on a fine grid in the
    specfied square. If the point being examined escapes, it is plotted in
    "black". Otherwise it is plotted in "white". The Julia set consists of the
    boundary between these two regions. The fractal nature of Julia sets may be
    explored with different centers and magnifications. The user should be
    warned that this proghram runs very slowly and for some functions, centers
    and magnifications, it may take days to produce a Julia set.

J - JULIA.Z*Z+C: This explores the classical function Z^2 + C. The user simply
    enters the real and imaginary parts of the complex constant C, a center
    point and a magnification. Then the Julia set of Z^2 + C is generated in
    the region chosen, s l o w l y...

K - JULIA.BACK.ORBITS: The Julia set of a function may be constructed by a
    faster but less accurate method using "back-orbits".. For an explanation of
    this method, see Devaney's book.

L - JULIA.Z*Z+C.BDRY.SCAN: The Julia set may be indicated more directly by
    trying to plot only those points on the boundary between those which escape
    and those which don't. The method is essentially the same as the one used
    in program I.

M - MANDELBROT: The Mandelbrot set is similar in principle to the Julia set.
    (Benoit Mandelbrot would have a stroke reading the preceeding sentence.)
    The function examined is F(Z) + C, but instead of finding the boundary
    between escaping and non-escaping Z, F(Z) + C is iterated for various
    values of C, starting with Z = 0. The Mandelbrot set is the boundary
    between those C which cause F(Z) + C to escape when iterated from Z = 0,
    and those values of C which result in finite orbits. The classic Mandelbrot
    set results from Z*Z + C, using center 0.75 and magnification 1.7 in this
    program. The whole set is between C = -2 and C = 0.25. Details of the
    Mandelbrot set may be explored by choosing other centers and higher
    magnifications.
    The magnification possible by any computer program which constructs Julia
    or Mandelbrot sets depends on the number of places to which the machine may
    accurately do arithmetic. With a normal Apple II, this is about 10 places,
    so magnifications of up to about 10E9 are possible. Beyond this, images
    tend to be too blocky and inaccurate.
    This program also runs very slowly, sometimes taking days to complete
    pictures.

N - MANDELBROT.BDRY.SCAN: Here the same boundary scanning technique is used as
    in program L. The result is a depiction of just the boundary between C
    which cause escape, annd C which result in finite orbits. This program runs
    slightly slower than the previous one.

O - SIERPINSKI: This is a famous fractal set constructed by dividing a triangle
    into four sub-triangles by connecting the midpoints of its sides, and
    removing the middle triangle. Divide each reamining triangle into four and
    again remove the middle one. Repeating this process endlessly results in
    the Sierpinski triangle.
    The Sierpinski triangle is generated in this program using a back-orbit
    process similar to that in program K. The process is relatively fast but
    innaccurate.
    The user is prompted for the coordinates of the vertices of the starting
    triangle. Try the screen vertices (140,0), (0,159), and (279,159).

P - SQUARE.MINUS.CROSS: This program is similar to the Sierpinski triangle. Use
    the points (0,0), (279,0), (0,159), and (279,159). This divides squares
    into 9 equal sub-squares and removes the middle ones including the center
    one (hence a cross).

Q - PENTAGON.MINUS.5-SPOKE: Again, this is like the prevoius two programs. Use
    (140,0), (0,50), (279,50), (95,159), and (185,159).

R - SIERPINSKI.SQUARE: Start with the same points as in program P. Unlike
    program P, the middle square is left.
    This, and the preceding three programs construct fractal sets of different
    Poincare dimensions. (Again, see Devaney's book for an explanation of
    Poincare dimension.)

S - ROTATING.4-CUBE: Several years ago, a mathematician at Brown University in
    Providence, R.I. made a film of a rotating 4-dimensional cube. Many
    pictures were taken of computer generated images of the 4-cube, each
    rotated a small amount from the previous image. When viewed together at a
    rate of 24 frames per second, the result was the rotating 4-cube. Here the
    computer draws a 4-cube projected on a 2-dimensional viewing plane very
    rapidly. The user may vary the attitude of the viewing plane and see the
    4-cube from other attitudes. The rotation controls are the A, B, C, D, and
    E keys. To move back, use the lower case key corresponding to one of the
    previous upper case keys.
    Due to a problem in an "orthonormalization" routine which uses an absolute
    value to ensure a quantity beneath a radical is always positive, there are
    some rotation steps which cause the cube to suddenly reflect about a plane
    in 4-space. This glitch has not been thoroughly investigated yet. But have
    fun playing with with a 4-dimensional cube!

T - GRAPH.Y-F(X).LO-RES.QUICK: is program is the low resolution counterpart of
    some of the previous programs. It is VERY rapid, but only has a resolution
    of 40 pixels hoizontally, and 40 vertically. It does have an automatic
    range-finding option. 

U - FOUR-D.HDN.PT.REMOVAL: "Vision" in 4-dimensional space is not as simple a
    matter as in 3-space. It is clear in 3-space what is meant by "in front of"
    or behind" because a "line" of sight may be dividedf in two parts (front
    and back) by a point. In 3-dimensions, the space at right angles
    (orthogonal) to a viewing plane is a (1-dimensional) line. However, in
    4-dimensional space, the subspace orthogonal to a plane is some other
    2-dimensional plane! It takes more than a single point to divide a plane
    into two pieces.
    If you have tried to graph even fairly simple functions in 4 dimensions,
    you have seen the extremely confusing plethora of super-imposed curves
    generated. In order to clean-up such pictures, a hiddenn point removal
    algorithm has been developed which searches a quarter plane to see if it
    passes through the object being graphed. If it does, the point at which the
    quarter-plane starts (at the object itself) is not graphed (hidden).
    Otherwise the point on the object is plotted.
    To search the quarter plane, points on a grid within that quarter-plane are
    tested, much the same as points along a half-line were tested in the 3-D
    hidden point removal algorithm used previously. Two search intervals are
    entered by the user instead of the one in the previous hidden-point removal
    algorithm 3-D graphing program. Much more time is taken to search the whole
    part of a quarter-plane within the domain-range region instead of just the
    previous piece of a half-line. This is done for each point being graphed
    (or hidden), so time really mounts up!
    This program is still experimental, so the choice of the quarter-plane
    remains in doubt. To try all 4 quarter-planes, you may use the four
    combinations of positive and negative search intervals: ++, +-, --, and -+.
    You may try to view the 4-dimensional paraboloid U = X*X + Y*Y + Z*Z. limit
    X, Y and Z to the unit intervals from -1 to 1, and use a range from 0 to 3.
    Search intervals of .2 each may be used, in each of the 4 possible sign
    assignments but remember, this program takes a very long time to run.

V - RIEMANN.SURFACE.HDN.PT.REMOVAL: This program attempts to do for Riemann
    surfaces what the previous program attempts for generalized 3-dimensional
    hyper-surfaces in 4-dimensional space.
    
    
    -END OF DOCUMENTATION FILE-