NeXT TypedStream Data | 1995-06-12 | 12.1 KB | 293 lines
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/usr/include/machine/vm_types.h
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.Mandelbrot Set of Quadratic Rational Functions
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r im:
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- The a-plane section, when b = r a + s, of
OtherViews
1 The (1/a)-plane section, when b = r a + s, of
- The b-plane section, when a = r b + s, of
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4 the Mandelbrot Set of F(z) = (1/a)(z + b + (1/z))
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2The Mandelbrot Set of Quadratic Rational Functions
Version 1.1 Dec 1992
by William Gilbert
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RatMan
wgilbert@fatou.uwaterloo.ca
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[2783c]{\rtf0\ansi{\fonttbl\f0\fswiss Helvetica;}
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\pard\tx520\tx1060\tx1600\tx2120\tx2660\tx3200\tx3720\tx4260\tx4800\tx5320\f0\b0\i0\ul0\fs24\fc0 The general quadratic rational function\
F(z) = (1/a)(z + b + (1/z))\
has two complex parameters a and b.\
The Mandelbrot set consists of the parameters (a,b), in 4 dimensional space, for which the filled-in Julia set is connected. This program illustrates 2 dimensional sections of this 4 dimensional set where\
b = ra + s (r, s constants).\
The values r = -1 and s = 2, yield the standard \
Mandelbrot set of quadratic polynomials, using the view shown on Plate 189 of Mandelbrot's book, since\
the function F(z) = (1/a)(z + b + (1/z)) is conjugate to the function bz(1 - z) when a + b = 2.\
===========================================\
\b Using the Program
\b0 \
There are various preset values of the parameters that give interesting sections. You can magnify a section of the view by dragging the mouse over the area and recomputing. When the program is running, the user interface is blocked except for the STOP button. If you stop the program before it finishes, you can continue by pressing CONT, assuming you have not changed any parameters.\
===========================================\
\b Printing
\b0 \
The program recomputes the view at a higher resolution when printing, so that it is possible to print at a very fine resolution. This could take up to 30 times as long as the screen viewing, so you should print to a file and then print the PostScript file, in order to avoid blocking the print queue.\
===========================================\
\b The Mathematics of the Program
\b0 \
The function F(z) has critical points at 1 and -1, and one of its three fixed points at infinity. Infinity is an attractive fixed point if and only if |a|<1.\
The program computes the iterates of the critical points of F(z) corresponding to each pixel. The pixel is colored black if the iterates of the critical points are not close and if they do not approach a common k-cycle, where k is less than or equal to the cycles parameter.\
===========================================\
\b References:
\b0 \
B. Mandelbrot, The Fractal Geometry of Nature,\
Freeman, N.Y. 1983.\
J. Milnor, Remarks on Quadratic Rational Maps\
(Preprint, SUNY at StonyBrook Institute for\
Mathematical Sciences, 1992)\
Yin Yongcheng, On the Julia Sets of Quadratic\
Rational Maps, \
Complex Variables, 18 (1992) pp 141-147.\
===========================================\
\b Written
\b0 July 1991 and modified December 1992\
by William Gilbert, Pure Mathematics Department\
University of Waterloo, Waterloo, Ontario, Canada
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University of Waterloo, Canada
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(.txt)