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Mandelbrot set: Difference between revisions

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Mandelbrot set (view source)

Revision as of 13:37, 16 July 2023

405 bytes added ,  11 months ago
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→‎Normal Map Effect, Mercator Projection and Deep Zoom Images: Much nicer stripes
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===Normal Map Effect, Mercator Projection and Deep Zoom Images===
The Mandelbrot set is represented by distance estimation and normal maps using NumPy and complex matrices (cf. Arnaud Chéritat: [https://www.math.univ-toulouse.fr/~cheritat/wiki-draw/index.php/Mandelbrot_set#Normal_map_effect ''Normal map effect'']). Note that the second derivative (ddZ) grows very fast, so the second method can only be used for small iteration numbers (n <= 400). See also [https://www.shadertoy.com/view/wtscDX Julia Stripes] on Shadertoy.
<syntaxhighlight lang="python">import numpy as np
import matplotlib.pyplot as plt
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direction, height = 45, 1.5 # direction and height of the incoming light
stripes, damping = 5.0, 2.0 # stripe density and damping parameter
v = np.exp(direction / 180 * np.pi * 1j) # unit 2D vector in this direction
 
x = np.linspace(0, 2, num=d+1)
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Z, dZ, ddZ = np.zeros_like(C), np.zeros_like(C), np.zeros_like(C)
D, S, T = np.zeros(C.shape), np.zeros(C.shape), np.zeros(C.shape)
 
for k in range(n):
M = Z.real ** 2 + Z.imag ** 2 < r ** 2
S[M], T[M] = S[M] + np.sin(stripes * np.angle(Z[M])), T[M] + 1
Z[M], dZ[M], ddZ[M] = Z[M] ** 2 + C[M], 2 * Z[M] * dZ[M] + 1, 2 * (dZ[M] ** 2 + Z[M] * ddZ[M])
 
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N = abs(Z) > 2 # normal map effect 1 (potential function)
P, Q = S[N] / T[N], (S[N] + np.sin(stripes * np.angle(Z[N]))) / (T[N] + 1)
F = 1 - np.log2(np.log(np.abs(Z[N])) / np.log(r))
H = P + (Q - P) * F * F * (3 - 2 * F) # hermite interpolation
U = Z[N] / dZ[N] # normal vectors to the equipotential lines
U, Sv = U / abs(U), 1 + np.sinexp(100direction / 180 * np.angle(U))pi /* 101j) # unit normal vectors and stripesunit 2D vector
TD[N] = np.maximum((U.real * v.real + U.imag * v.imag + S(1 + H / damping) * height) / (1 + height), 0)
 
plt.imshow(TD ** 1.0, cmap=plt.cm.bone, origin="lower")
plt.savefig("Mandelbrot_normal_map_1.png", dpi=200)
 
N = abs(Z) > 2 # normal map effect 2 (distance estimation)
U = Z[N] * dZ[N] * ((1 + np.log(abs(Z[N]))) * np.conj(dZ[N] ** 2) - np.log(abs(Z[N])) * np.conj(Z[N] * ddZ[N]))
U, v = U / abs(U), np.exp(direction / 180 * np.pi * 1j) # unit normal vectors toand theunit equidistant2D linesvector
TD[N] = np.maximum((U.real * v.real + U.imag * v.imag + height) / (1 + height), 0)
 
plt.imshow(TD ** 1.0, cmap=plt.cm.afmhot, origin="lower")
plt.savefig("Mandelbrot_normal_map_2.png", dpi=200)</syntaxhighlight>
 
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