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r""" 

Mandelbrot and Julia sets (Cython helper) 

  

This is the helper file providing functionality for mandel_julia.py. 

  

AUTHORS: 

  

- Ben Barros 

  

""" 

  

#***************************************************************************** 

# Copyright (C) 2017 BEN BARROS <bbarros@slu.edu> 

# 

# This program is free software: you can redistribute it and/or modify 

# it under the terms of the GNU General Public License as published by 

# the Free Software Foundation, either version 2 of the License, or 

# (at your option) any later version. 

# http://www.gnu.org/licenses/ 

#***************************************************************************** 

  

from __future__ import absolute_import, division 

from sage.plot.colors import Color 

from sage.repl.image import Image 

from copy import copy 

from cysignals.signals cimport sig_check 

from sage.rings.complex_field import ComplexField 

from sage.functions.log import exp, log 

from sage.symbolic.constants import pi 

  

def fast_mandelbrot_plot(double x_center, double y_center, double image_width, 

long max_iteration, long pixel_count, long level_sep, long color_num, base_color): 

  

r""" 

Plots the Mandelbrot set in the complex plane for the map `Q_c(z) = z^2 + c`. 

  

INPUT: 

  

- ``x_center`` -- double, real part of the center point in the complex plane. 

  

- ``y_center`` -- double, imaginary part of the center point in the complex plane. 

  

- ``image_width`` -- double, width of the image in the complex plane. 

  

- ``max_iteration`` -- long, maximum number of iterations the map `Q_c(z)` considered. 

  

- ``pixel_count`` -- long, side length of image in number of pixels. 

  

- ``level_sep`` -- long, number of iterations between each color level. 

  

- ``color_num`` -- long, number of colors used to plot image. 

  

- ``base_color`` -- list, RGB color used to determine the coloring of set. 

  

OUTPUT: 

  

24-bit RGB image of the Mandelbrot set in the complex plane 

  

EXAMPLES: 

  

Plot the Mandelbrot set with the center point `-1 + 0i`:: 

  

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import fast_mandelbrot_plot 

sage: fast_mandelbrot_plot(-1, 0, 4, 500, 600, 1, 20, [40, 40, 40]) # long time 

600x600px 24-bit RGB image 

  

We can focus on smaller parts of the set by adjusting image_width:: 

  

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import fast_mandelbrot_plot 

sage: fast_mandelbrot_plot(-1.11, 0.2283, 1/128, 2000, 500, 1, 500, [40, 100, 100]) # long time 

500x500px 24-bit RGB image 

""" 

  

cdef long i, j, col, row, level, color_value, iteration 

cdef double k, x_corner, y_corner, step_size, x_coor, y_coor, new_x, new_y 

cdef M, pixel, color_list 

  

# Make sure image_width is positive 

image_width = abs(image_width) 

  

# Initialize an image to the color black and access the pixels 

M = Image("RGB", (pixel_count,pixel_count), 'black') 

pixel = M.pixels() 

  

# Take the given base color and create a list of evenly spaced 

# colors between the given base color and white. The number of 

# colors in the list depends on the variable color_num. 

if type(base_color) == Color: 

# Convert Color to RGB list 

base_color = [int(k*255) for k in base_color] 

color_list = [] 

for i in range(color_num): 

sig_check() 

color_list.append(copy(base_color)) 

for j in range(3): 

color_list[i][j] += i * (255 - color_list[i][j]) // color_num 

color_list[i] = tuple(color_list[i]) 

  

# First, we determine the complex coordinates of the point in the top left 

# corner of the image. Then, we loop through each pixel in the image and 

# assign it complex coordinates relative to the image's top left corner. 

x_corner = x_center - image_width/2 

y_corner = y_center + image_width/2 

step_size = image_width / pixel_count 

for col in range(pixel_count): 

x_coor = x_corner + col*step_size 

for row in range(pixel_count): 

sig_check() 

y_coor = y_corner - row*step_size 

  

# We compute the orbit of 0 under the map Q(z) = z^2 + c 

# until we either reach the maximum number of iterations 

# or find a point in the orbit with modulus greater than 2 

new_x, new_y = 0.0, 0.0 

iteration = 0 

while (new_x**2 + new_y**2 <= 4.0 and iteration < max_iteration): 

sig_check() 

new_x, new_y = new_x**2 - new_y**2 + x_coor, \ 

2*new_x*new_y + y_coor 

iteration += 1 

  

# If the point escapes to infinity, assign the point a color 

# based on how fast it escapes. The more iterations it takes for 

# a point to escape to infinity, the lighter its color will be. 

# Otherwise, assume the point is in the Mandelbrot set and leave 

# it black. 

if iteration != max_iteration: 

# Assign each point a level based on its number of iterations. 

level = iteration // level_sep 

# Assign the pixel a color based on it's level. If we run out 

# of colors, assign it the last color in the list. 

if level < color_num: 

pixel[col,row] = color_list[level] 

else: 

pixel[col,row] = color_list[-1] 

return M 

  

cpdef fast_external_ray(double theta, long D=30, long S=10, long R=100, 

long pixel_count=500, double image_width=4, long prec=300): 

r""" 

Returns a list of points that approximate the external ray for a given angle. 

  

INPUT: 

  

- ``theta`` -- double, angle between 0 and 1 inclusive. 

  

- ``D`` -- long (optional - default: ``25``) depth of the approximation. As ``D`` increases, the external ray gets closer to the boundary of the Mandelbrot set. 

  

- ``S`` -- long (optional - default: ``10``) sharpness of the approximation. Adjusts the number of points used to approximate the external ray (number of points is equal to ``S*D``). 

  

- ``R`` -- long (optional - default: ``100``) radial parameter. If ``R`` is sufficiently large, the external ray reaches enough close to infinity. 

  

- ``pixel_count`` -- long (optional - default: ``500``) side length of image in number of pixels. 

  

- ``image_width`` -- double (optional - default: ``4``) width of the image in the complex plane. 

  

- ``prec`` -- long (optional - default: ``300``) specifies the bits of precision used by the Complex Field when using Newton's method to compute points on the external ray. 

  

OUTPUT: 

  

List of tuples of Real Interval Field Elements 

  

EXAMPLES:: 

  

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import fast_external_ray 

sage: fast_external_ray(0,S=1,D=1) 

[(100.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000, 

0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000), 

(9.51254777713729174697578576623132297117784691109499464854806785133621315075854778426714908, 

0.000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000)] 

  

  

:: 

  

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import fast_external_ray 

sage: fast_external_ray(1/3,S=1,D=1) 

[(-49.9999999999999786837179271969944238662719726562500000000000000000000000000000000000000000, 

86.6025403784438765342201804742217063903808593750000000000000000000000000000000000000000000), 

(-5.50628047023173006234970878097113901879832542655926629309001652388544528575532346900138516, 

8.64947510053972513843999918917106032664030380426885745306040284140385975750462108180377187)] 

  

:: 

  

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import fast_external_ray 

sage: fast_external_ray(0.75234,S=1,D=1) 

[(1.47021239172637052661229972727596759796142578125000000000000000000000000000000000000000000, 

-99.9891917935294287644865107722580432891845703125000000000000000000000000000000000000000000), 

(-0.352790406744857508500937144524776555433184352559852962308757189778284058275081335121601384, 

-9.98646630765023514178761177926164047797465369576787921409326037870837930920646860774032363)] 

""" 

  

cdef: 

CF = ComplexField(prec) 

PI = CF.pi() 

I = CF.gen() 

c_0, r_m, t_m, temp_c, C_k, D_k, old_c, x, y, dist 

int k, j, t 

double difference, m 

double error = pixel_count * 0.0001 

  

double pixel_width = image_width / pixel_count 

  

# initialize list with c_0 

c_list = [CF(R*exp(2*PI*I*theta))] 

  

# Loop through each subinterval and approximate point on external ray. 

for k in range(1,D+1): 

for j in range(1,S+1): 

m = (k-1)*S + j 

r_m = CF(R**(2**(-m/S))) 

t_m = CF(r_m**(2**k) * exp(2*PI*I*theta * 2**k)) 

temp_c = c_list[-1] 

difference = error 

  

# Repeat Newton's method until points are close together. 

while error <= difference: 

sig_check() 

old_c = temp_c 

# Recursive formula for iterates of q(z) = z^2 + c 

C_k, D_k = CF(old_c), CF(1) 

for t in range(k): 

C_k, D_k = C_k**2 + old_c, CF(2)*D_k*C_k + CF(1) 

temp_c = old_c - (C_k - t_m) / D_k # Newton map 

difference = abs(old_c) - abs(temp_c) 

  

dist = (2*C_k.abs()*(C_k.abs()).log()) / D_k.abs() 

if dist < pixel_width: 

break 

c_list.append(CF(temp_c)) 

if dist < pixel_width: 

break 

  

# Convert Complex Field elements into tuples. 

for k in range(len(c_list)): 

x,y = c_list[k].real(), c_list[k].imag() 

c_list[k] = (x, y) 

  

return c_list 

  

cpdef convert_to_pixels(point_list, double x_0, double y_0, double width, 

long number_of_pixels): 

r""" 

Converts cartesian coordinates to pixels within a specified window. 

  

INPUT: 

  

- ``point_list`` -- list of tuples, points in cartesian coordinates. 

  

- ``x_0`` -- double, x-coordinate of the center of the image. 

  

- ``y_0`` -- double, y-coordinate of the center of the image. 

  

- ``width`` -- double, width of visible window in caresian coordinates. 

  

- ``number_of_pixels`` -- long, width of image in pixels. 

  

OUTPUT: 

  

List of tuples of integers representing pixels. 

  

EXAMPLES:: 

  

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import convert_to_pixels 

sage: convert_to_pixels([(-1,3),(0,-4),(5,0)], 0, 0, 12, 100) 

[(42, 25), (50, 83), (92, 50)] 

""" 

cdef: 

k, pixel_list, x_corner, y_corner, step_size 

long x_pixel, y_pixel 

pixel_list = [] 

  

# Compute top left corner of window and step size 

x_corner = x_0 - width/2 

y_corner = y_0 + width/2 

step_size = number_of_pixels / width 

  

# Convert each point in list to pixel coordinates 

for k in point_list: 

sig_check() 

x_pixel = round((k[0] - x_corner) * step_size) 

y_pixel = round((y_corner - k[1]) * step_size) 

pixel_list.append((x_pixel, y_pixel)) 

return pixel_list 

  

cpdef get_line(start, end): 

r""" 

Produces a list of pixel coordinates approximating a line from a starting 

point to an ending point using the Bresenham's Line Algorithm. 

  

REFERENCE: 

  

[Br2016]_ 

  

INPUT: 

  

- ``start`` -- tuple, starting point of line. 

  

- ``end`` -- tuple, ending point of line. 

  

OUTPUT: 

  

List of tuples of integers approximating the line between two pixels. 

  

EXAMPLES:: 

  

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import get_line 

sage: get_line((0, 0), (3, 4)) 

[(0, 0), (1, 1), (1, 2), (2, 3), (3, 4)] 

  

:: 

  

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import get_line 

sage: get_line((3, 4), (0, 0)) 

[(3, 4), (2, 3), (1, 2), (1, 1), (0, 0)] 

""" 

# Setup initial conditions 

cdef: 

long x1, x2, y1, y2, dx, dy, error, ystep, y 

is_steep, swapped, points 

x1, y1 = start 

x2, y2 = end 

dx, dy = x2 - x1, y2 - y1 

  

# Determine how steep the line is 

is_steep = abs(dy) > abs(dx) 

  

# Rotate line 

if is_steep: 

x1, y1 = y1, x1 

x2, y2 = y2, x2 

  

# Swap start and end points if necessary and store swap state 

swapped = False 

if x1 > x2: 

x1, x2 = x2, x1 

y1, y2 = y2, y1 

swapped = True 

  

# Recalculate differentials 

dx, dy = x2 - x1, y2 - y1 

  

# Calculate error 

error = int(dx / 2.0) 

ystep = 1 if y1 < y2 else -1 

  

# Iterate over bounding box generating points between start and end 

y = y1 

points = [] 

for x in range(x1, x2 + 1): 

sig_check() 

coord = (y, x) if is_steep else (x, y) 

points.append(coord) 

error -= abs(dy) 

if error < 0: 

y += ystep 

error += dx 

  

# Reverse the list if the coordinates were swapped 

if swapped: 

points.reverse() 

return points 

  

cpdef fast_julia_plot(double c_real, double c_imag, 

double x_center=0, double y_center=0, double image_width=4, 

long max_iteration=500, long pixel_count=500, long level_sep=2, 

long color_num=40, base_color=[50, 50, 50]): 

r""" 

Plots the Julia set for a given `c` value in the complex plane for the map `Q_c(z) = z^2 + c`. 

  

INPUT: 

  

- ``c_real`` -- double, Real part of `c` value that determines Julia set. 

  

- ``c_imag`` -- double, Imaginary part of `c` value that determines Julia set. 

  

- ``x_center`` -- double (optional - default: ``0.0``), Real part of center point. 

  

- ``y_center`` -- double (optional - default: ``0.0``), Imaginary part of center point. 

  

- ``image_width`` -- double (optional - default: ``4.0``), width of image in the complex plane. 

  

- ``max_iteration`` -- long (optional - default: ``500``), maximum number of iterations the map ``Q_c(z)``. 

  

- ``pixel_count`` -- long (optional - default: ``500``), side length of image in number of pixels. 

  

- ``level_sep`` -- long (optional - default: ``2``), number of iterations between each color level. 

  

- ``color_num`` -- long (optional - default: ``40``), number of colors used to plot image. 

  

- ``base_color`` -- RGB color (optional - default: ``[50, 50, 50]``), color used to determine the coloring of set. 

  

OUTPUT: 

  

24-bit RGB image of the Julia set in the complex plane. 

  

EXAMPLES: 

  

Plot the Julia set for `c=-1+0i`:: 

  

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import fast_julia_plot 

sage: fast_julia_plot(-1, 0) 

500x500px 24-bit RGB image 

  

We can focus on smaller parts of the set by adjusting ``image_width``:: 

  

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import fast_julia_plot 

sage: fast_julia_plot(-0.7, 0.3, x_center=.75, image_width=0.01, color_num=100) 

500x500px 24-bit RGB image 

""" 

  

cdef long i, j, col, row, level, color_value, iteration 

cdef double k, x_corner, y_corner, step_size, x_coor, y_coor, new_x, new_y 

cdef M, pixel, color_list 

  

# Make sure image_width is positive 

image_width = abs(image_width) 

  

# Initialize an image to the color black and access the pixels 

J = Image("RGB", (pixel_count,pixel_count), 'black') 

Jp = J.pixels() 

  

# Take the given base color and create a list of evenly spaced 

# colors between the given base color and white. The number of 

# colors in the list depends on the variable color_num. 

if type(base_color) == Color: 

# Convert Color to RGB list 

base_color = [int(k*255) for k in base_color] 

color_list = [] 

for i in range(color_num): 

sig_check() 

color_list.append(copy(base_color)) 

for j in range(3): 

color_list[i][j] += i * (255 - color_list[i][j]) // color_num 

color_list[i] = tuple(color_list[i]) 

  

# First, we determine the complex coordinates of the point in the top left 

# corner of the image. Then, we loop through each pixel in the image and 

# assign it complex coordinates relative to the image's top left corner. 

x_corner = x_center - image_width/2 

y_corner = y_center + image_width/2 

step_size = image_width / pixel_count 

for col in range(pixel_count): 

x_coor = x_corner + col*step_size 

for row in range(pixel_count): 

sig_check() 

y_coor = y_corner - row*step_size 

  

# We compute the orbit of each pixel under the map Q(z) = z^2 + c 

# until we either reach the maximum number of iterations 

# or find a point in the orbit with modulus greater than 2 

new_x, new_y = x_coor, y_coor 

iteration = 0 

while (new_x**2 + new_y**2 <= 4.0 and iteration < max_iteration): 

sig_check() 

new_x, new_y = new_x**2 - new_y**2 + c_real, \ 

2*new_x*new_y + c_imag 

iteration += 1 

  

# If the point escapes to infinity, assign the point a color 

# based on how fast it escapes. The more iterations it takes for 

# a point to escape to infinity, the lighter its color will be. 

# Otherwise, assume the point is in the Julia set and leave 

# it black. 

if iteration != max_iteration: 

# Assign each point a level based on its number of iterations. 

level = iteration // level_sep 

# Assign the pixel a color based on it's level. If we run out 

# of colors, assign it the last color in the list. 

if level < color_num: 

Jp[col,row] = color_list[level] 

else: 

Jp[col,row] = color_list[-1] 

  

return J 

  

cpdef julia_helper(double c_real, double c_imag, double x_center=0, 

double y_center=0, double image_width=4, long max_iteration=500, 

long pixel_count=500, long level_sep=2, long color_num=40, 

base_color=[50, 50, 50], point_color=[255, 0, 0]): 

  

r""" 

Helper function that returns the image of a Julia set for a given 

`c` value side by side with the Mandelbrot set with a point denoting 

the `c` value. 

  

INPUT: 

  

- ``c_real`` -- double, Real part of `c` value that determines Julia set. 

  

- ``c_imag`` -- double, Imaginary part of `c` value that determines Julia set. 

  

- ``x_center`` -- double (optional - default: ``0.0``), Real part of center point. 

  

- ``y_center`` -- double (optional - default: ``0.0``), Imaginary part of center point. 

  

- ``image_width`` -- double (optional - default: ``4.0``), width of image in the complex plane. 

  

- ``max_iteration`` -- long (optional - default: ``500``), maximum number of iterations the map ``Q_c(z)``. 

  

- ``pixel_count`` -- long (optional - default: ``500``), side length of image in number of pixels. 

  

- ``level_sep`` -- long (optional - default: ``2``), number of iterations between each color level. 

  

- ``color_num`` -- long (optional - default: ``40``), number of colors used to plot image. 

  

- ``base_color`` -- RGB color (optional - default: ``[50, 50, 50]``), color used to determine the coloring of set. 

  

- ``point_color`` -- RGB color (optional - default: ``[255, 0, 0]``), color of the point `c` in the Mandelbrot set. 

  

OUTPUT: 

  

24-bit RGB image of the Julia and Mandelbrot sets in the complex plane. 

  

EXAMPLES: 

  

Plot the Julia set for `c=-1+0i`:: 

  

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import julia_helper 

sage: julia_helper(-1,0) 

1001x500px 24-bit RGB image 

  

We can focus on smaller parts of the set by adjusting ``image_width``:: 

  

sage: from sage.dynamics.complex_dynamics.mandel_julia_helper import julia_helper 

sage: julia_helper(-.5, .6, y_center=0.178, image_width=0.01) # long time 

1001x500px 24-bit RGB image 

""" 

  

cdef: 

int i, j 

M, Mp, G, Gp, J, Jp, CP 

  

# Initialize the Julia set 

J = fast_julia_plot(c_real, c_imag, x_center, y_center, image_width, 

max_iteration, pixel_count, level_sep, color_num, base_color) 

Jp = J.pixels() 

  

# Initialize the image with Julia set on left side 

# Add white border between images 

G = Image("RGB", (2*pixel_count+1,pixel_count), 'white') 

Gp = G.pixels() 

for i in range(pixel_count): 

for j in range(pixel_count): 

Gp[i,j] = Jp[i,j] 

  

# Plot the Mandelbrot set on the right side 

M = fast_mandelbrot_plot(-1, 0, 4, 500, pixel_count, 1, 30, base_color) 

Mp = M.pixels() 

for i in range(pixel_count+1,2*pixel_count): 

for j in range(pixel_count): 

Gp[i,j] = Mp[int(i-pixel_count),j] 

  

# Convert Color to RGB list if necessary 

if type(point_color) == Color: 

point_color = [int(k*255) for k in point_color] 

  

# Add a cross representing c-value to the Mandelbrot set. 

CP = convert_to_pixels([(c_real, c_imag)], -1, 0, 4, pixel_count) 

for i in range(-3,4): 

# Loop through x and y coordinates and check if they are in image 

if min(CP[0][0]+i, CP[0][1]) >= 0 and \ 

max(CP[0][0]+i, CP[0][1]) < pixel_count: 

Gp[CP[0][0]+i+pixel_count+1, CP[0][1]] = tuple(point_color) 

if min(CP[0][0], CP[0][1]+i) >= 0 and \ 

max(CP[0][0], CP[0][1]+i) < pixel_count: 

Gp[CP[0][0]+pixel_count+1, CP[0][1]+i] = tuple(point_color) 

  

return G