58 lines
1.7 KiB
Python
58 lines
1.7 KiB
Python
import math
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from PIL import Image
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width, height = 800, 800
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max_iter = 40
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tolerance = 0.001 # Wie nah müssen wir an der Nullstelle sein?
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# Die drei Nullstellen von z^3 - 1 = 0
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roots = [
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complex(1, 0),
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complex(-0.5, math.sqrt(3) / 2),
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complex(-0.5, -math.sqrt(3) / 2)
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]
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# Farben für die drei Nullstellen (Rot, Grün, Blau)
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root_colors = [
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(255, 0, 0),
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(0, 255, 0),
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(0, 0, 255)
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]
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img = Image.new('RGB', (width, height), (0, 0, 0))
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pixels = img.load()
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for py in range(height):
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for px in range(width):
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# Koordinaten umrechnen (von -2 bis 2)
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zx = -2.0 + (px / width) * 4.0
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zy = -2.0 + (py / height) * 4.0
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z = complex(zx, zy)
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found = False
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for n in range(max_iter):
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if abs(z) < 0.000001: break # Vermeidung von Division durch Null
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# Newton-Schritt für f(z) = z^3 - 1
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# Formel: z = z - f(z) / f'(z) => z = z - (z^3 - 1) / (3 * z^2)
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z_next = z - (z ** 3 - 1) / (3 * z ** 2)
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# Prüfen, ob wir nah genug an einer der Wurzeln sind
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for i, root in enumerate(roots):
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diff = z_next - root
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if abs(diff) < tolerance:
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# Farbe basierend auf der Wurzel, Helligkeit basierend auf Iterationen
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brightness = int(255 * (1 - n / max_iter))
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r, g, b = root_colors[i]
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pixels[px, py] = (
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int(r * (brightness / 255)),
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int(g * (brightness / 255)),
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int(b * (brightness / 255))
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)
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found = True
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break
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if found: break
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z = z_next
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img.save("newton_fractal.png")
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img.show() |