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julia_set.py
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julia_set.py
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"""
A script to generate an animated gif of the Julia set.
The code for the JULIA SET 2 is taken from the Matplotlib documentation
website: https://matplotlib.org/matplotblog/posts/animated-fractals/?ssp=1&darkschemeovr=1&setlang=de-DE&safesearch=moderate
author: Fabrizio Musacchio (fabriziomusacchio.com)
date: Aug 2, 2021
"""
# %% IMPORTS
import numpy as np
import matplotlib.pyplot as plt
from matplotlib.animation import FuncAnimation
# %% JULIA SET 1:
def julia_set(c, width, height, x_min, x_max, y_min, y_max, max_iter):
x = np.linspace(x_min, x_max, width)
y = np.linspace(y_min, y_max, height)
X, Y = np.meshgrid(x, y)
Z = X + 1j * Y
img = np.zeros_like(Z, dtype=int)
for i in range(max_iter):
mask = np.abs(Z) < 4
Z[mask] = Z[mask] ** 2 + c
img += mask
return img
# set up the figure and subplots:
fig, (ax1, ax2) = plt.subplots(1, 2, figsize=(10, 5))
# common parameters:
width, height = 400, 400
x_min, x_max = -2.0, 2.0
y_min, y_max = -2.0, 2.0
c_values = np.linspace(-0.8, 0.8, 100) # different values of the parameter 'c'
max_iter = 100
# plot 1 - full Julia set:
img1 = ax1.imshow(julia_set(c_values[0], width, height, x_min, x_max, y_min, y_max, max_iter),
cmap='RdPu', extent=(x_min, x_max, y_min, y_max), origin='lower')
ax1.set_title('Julia Set (Full)')
ax1.set_xlabel('Re(c)')
ax1.set_ylabel('Im(c)')
# plot 2 - zoomed-in Julia set:
zoom_factor = 0.5
zoom_x_min = x_max - (x_max - x_min) * zoom_factor
zoom_x_max = x_max
zoom_y_min = y_max - (y_max - y_min) * zoom_factor
zoom_y_max = y_max
img2 = ax2.imshow(julia_set(c_values[0], width, height, zoom_x_min, zoom_x_max, zoom_y_min, zoom_y_max, max_iter),
cmap='RdPu', extent=(zoom_x_min, zoom_x_max, zoom_y_min, zoom_y_max), origin='lower')
ax2.set_title('Julia Set (Zoomed)')
ax2.set_xlabel('Re(c)')
ax2.set_ylabel('Im(c)')
# update function:
def update(frame):
# update plot 1 - full Julia set:
img1.set_array(julia_set(c_values[frame], width, height, x_min, x_max, y_min, y_max, max_iter))
ax1.set_title(f"c = {c_values[frame]:.2f}")
# update plot 2 - zoomed-in Julia set:
img2.set_array(julia_set(c_values[frame], width, height, zoom_x_min, zoom_x_max, zoom_y_min, zoom_y_max, max_iter))
ax2.set_title(f"c = {c_values[frame]:.2f}")
# create and save the animation:
anim = FuncAnimation(fig, update, frames=len(c_values), interval=100)
anim.save('images/julia_set_animation.gif', writer='pillow', dpi=120)
plt.show()
# %% JULIA SET 2:
# define parameters:
x_start, y_start = -2, -2 # an interesting region starts here
width, height = 4, 4 # for 4 units up and right
density_per_unit = 200 # how many pixles per unit
# real and imaginary axis:
re = np.linspace(x_start, x_start + width, width * density_per_unit )
im = np.linspace(y_start, y_start + height, height * density_per_unit)
threshold = 20 # max allowed iterations
frames = 100 # number of frames in the animation
# we represent c as c = r*cos(a) + i*r*sin(a) = r*e^{i*a}:
r = 0.7885
a = np.linspace(0, 2*np.pi, frames)
fig = plt.figure(figsize=(10, 10)) # instantiate a figure to draw
ax = plt.axes() # create an axes object
def julia_quadratic(zx, zy, cx, cy, threshold):
"""Calculates whether the number z[0] = zx + i*zy with a constant c = x + i*y
belongs to the Julia set. In order to belong, the sequence
z[i + 1] = z[i]**2 + c, must not diverge after 'threshold' number of steps.
The sequence diverges if the absolute value of z[i+1] is greater than 4.
:param float zx: the x component of z[0]
:param float zy: the y component of z[0]
:param float cx: the x component of the constant c
:param float cy: the y component of the constant c
:param int threshold: the number of iterations to considered it converged
"""
# initial conditions
z = complex(zx, zy)
c = complex(cx, cy)
for i in range(threshold):
z = z**2 + c
if abs(z) > 4.: # it diverged
return i
return threshold - 1 # it didn't diverge
def animate(i):
ax.clear() # clear axes object
ax.set_xticks([], []) # clear x-axis ticks
ax.set_yticks([], []) # clear y-axis ticks
X = np.empty((len(re), len(im))) # the initial array-like image
cx, cy = r * np.cos(a[i]), r * np.sin(a[i]) # the initial c number
# iterations for the given threshold
for i in range(len(re)):
for j in range(len(im)):
X[i, j] = julia_quadratic(re[i], im[j], cx, cy, threshold)
img = ax.imshow(X.T, interpolation="bicubic", cmap='magma')
plt.tight_layout()
return [img]
anim = animation.FuncAnimation(fig, animate, frames=frames, interval=50, blit=True)
anim.save('images/julia_set_animation_2.gif', writer='imagemagick')