85 lines
4.5 KiB
Plaintext
85 lines
4.5 KiB
Plaintext
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MATHEMATICS OF THE FRACTAL TYPES
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Fractal Type(s) Formula(s) used
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----------------------- ---------------------------------------------
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Mandel, Julia Z(n+1) = Z(n)^2 + C
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Newton, Newtbasin (roots of) Z^n - 1, wnere n is an integer
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ComplexNewton, ComplexBasin (roots of) Z^a - b, where a,b are complex
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plasma (see the Plasma section for details)
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Mandelsine, Lambdasine Z(n+1) = Lambda * sine(Z(n)) + C
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Mandelcos, Lambdacos Z(n+1) = Lambda * cos(Z(n)) + C
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Mandelexp, Lambdaexp Z(n+1) = Lambda * exp(Z(n)) + C
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Mandelsinh, Lambdasinh Z(n+1) = Lambda * sinh(Z(n)) + C
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Mandelcosh, Lambdacosh Z(n+1) = Lambda * cosh(Z(n)) + C
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BarnsleyM1, BarnsleyJ1 Z(n+1) = (Z(n)-1) * C if Real(z) >= 0
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else (Z(n)+1) * modulus(C)/C
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BarnsleyM2, BarnsleyJ2 Z(n+1) = (Z(n)-1) * C if Real(Z(n))*Imag(C)
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+Real(C)*Imag(Z(n)) >= 0
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else (Z(n)+1) * C
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BarnsleyM3, BarnsleyJ3 Z(n+1) = (Real(Z(n))^2 - Imag(Z(n))^2 - 1)
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+ i * (2 * Real(Z((n)) * Imag(Z((n)))
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if Real(Z(n) > 0
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else (Real(Z(n))^2 - Imag(Z(n))^2 - 1
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+ lambda * Real(Z(n))
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+ i * (2 * Real(Z((n)) * Imag(Z((n))
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+ lambda * Real(Z(n))
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Sierpinski Z(n+1) = (2x, 2y - 1) if y > .5
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else (2x - 1, 2y) if x > .5
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else (2X, 2y)
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MandelLambda, Lambda Z(n+1) = (C) * (Z(n)^2) + C
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MarksMandel, MarksJulia Z(n+1) = (C^(Period-1)) * (Z(n)^2) + C
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("Period" is a parameter)
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Unity (see the Unity section for details)
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ifs, ifs3D (see the IFS section for details)
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Mandel4, Julia4 Z(n+1) = Z(n)^4 + C
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Test (as distributed, as simple Mandelbrot fractal)
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Mansinzsqrd, Julsinzsqrd Z(n+1) = Z(n)^2 + sin(Z(n)) + C
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Manzpower, Julzpower Z(n+1) = Z(n)^M + C (M is a parameter)
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Manzzpwr, Julzzpwr Z(n+1) = Z(n)^Z(n) + Z(n)^M + C
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Mansinexp, Julsinexp Z(n+1) = sin(Z(n)) + e^(Z(n)) + C
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popcorn Z(n+1) = x(n+1) + i * y(n+1), where
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x(n+1) = x(n) - 0.05*sin(y(n)) + tan(3*y(n))
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y(n+1) = y(n) - 0.05*sin(x(n)) + tan(3*x(n))
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demm, demj (Mandelbrot, Julia fractals calculated and
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colored using the "Distance Estimator" method
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Bifurcation (see the Bifurcation section for details)
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Lorenz, Lorenz3d Lorenz Attractor - orbits of differential
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equation
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x = x + (-a * x * dt) + (a * y * dt)
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y = y + (b * x * dt) - (y * dt) - (z * x * dt)
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z = z + (-c * z * dt) + (x * y * dt)
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(defaults: dt = .02, a = 5, b = 15, c = 1)
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(Lorenz3D is the Lorenz Attractor with the
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addition of 3D perspective transformations
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as specified by the IFS <E>ditor's
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transformation option)
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The following trig identities are invaluable for coding fractals that
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use complex-valued transcendental functions.
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e^(x+iy) = (e^x)cos(y) + i(e^x)sin(y)
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sin(x+iy) = sin(x)cosh(y) + icos(x)sinh(y)
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cos(x+iy) = cos(x)cosh(y) - isin(x)sinh(y)
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sinh(x+iy) = sinh(x)cos(y) + icosh(x)sin(y)
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cosh(x+iy) = cosh(x)cos(y) + isinh(x)sin(y)
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ln(x+iy) = (1/2)ln(x*x + y*y) + i(arctan(y/x) + 2kPi)
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(k = 0, +-1, +-2, +-....)
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sin(2x) sinh(2y)
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tan(x+iy) = ------------------ + i------------------
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cos(2x) + cosh(2y) cos(2x) + cosh(2y)
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sinh(2x) sin(2y)
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tanh(x+iy) = ------------------ + i------------------
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cosh(2x) + cos(2y) cosh(2x) + cos(2y)
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z^z = e^(log(z)*z)
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log(x + iy) = 1/2(log(x*x + y*y) + i(arc_tan(y/x))
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e^(x + iy) = (cosh(x) + sinh(x)) * (cos(y) + isin(y))
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= e^x * (cos(y) + isin(y))
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= (e^x * cos(y)) + i(e^x * sin(y))
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Extract from FRACTINT.DOC
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