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Improve Mandelbrot/Julia interact
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@jdemeyer
jdemeyer committed May 16, 2018
1 parent f25a0e8 commit 4de168d
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228 changes: 104 additions & 124 deletions src/sage/dynamics/complex_dynamics/mandel_julia.py
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Original file line number Diff line number Diff line change
Expand Up @@ -36,22 +36,27 @@
fast_julia_plot,
julia_helper)
from sage.dynamics.arithmetic_dynamics.generic_ds import DynamicalSystem
from sage.repl.ipython_kernel.interact import interact
from sage.plot.colors import Color
from sage.repl.image import Image
from sage.functions.log import logb
from sage.rings.rational_field import QQ
from sage.rings.all import CC
from sage.rings.all import QQ, CC
from sage.rings.polynomial.polynomial_ring_constructor import PolynomialRing
from sage.schemes.projective.projective_space import ProjectiveSpace
from sage.misc.prandom import randint
from sage.functions.other import floor


EPS = 0.00001


def mandelbrot_plot(**kwds):
def mandelbrot_plot(x_center=-1.0,
y_center=0.0,
image_width=4.0,
max_iteration=500,
pixel_count=500,
base_color='steelblue',
iteration_level=1,
number_of_colors=30,
interact=False):
r"""
Interactive plot of the Mandelbrot set for the map `Q_c(z) = z^2 + c`.
Expand All @@ -70,7 +75,7 @@ def mandelbrot_plot(**kwds):
[Dev2005]_
kwds:
INPUT:
- ``x_center`` -- double (optional - default: ``-1.0``), Real part
of center point.
Expand Down Expand Up @@ -107,77 +112,62 @@ def mandelbrot_plot(**kwds):
::
sage: mandelbrot_plot() # long time
sage: mandelbrot_plot()
500x500px 24-bit RGB image
::
sage: mandelbrot_plot(pixel_count=1000) # long time
sage: mandelbrot_plot(pixel_count=1000)
1000x1000px 24-bit RGB image
::
sage: mandelbrot_plot(x_center=-1.11, y_center=0.2283, image_width=1/128, # long time
sage: mandelbrot_plot(x_center=-1.11, y_center=0.2283, image_width=1/128,
....: max_iteration=2000, number_of_colors=500, base_color=[40, 100, 100])
500x500px 24-bit RGB image
To display an interactive plot of the Mandelbrot set in the Jupyter
Notebook, set ``interact`` to ``True``::
sage: mandelbrot_plot(interact=True)
Interactive function ... with 8 widgets...
interactive(children=(FloatSlider(value=-1.0, description=u'Real center'...
::
sage: mandelbrot_plot(interact=True, x_center=-0.75, y_center=0.25,
....: image_width=1/2, number_of_colors=75)
Interactive function ... with 8 widgets...
....: image_width=1/2, number_of_colors=75)
interactive(children=(FloatSlider(value=-0.75, description=u'Real center'...
"""
x_center = kwds.pop("x_center", -1.0)
y_center = kwds.pop("y_center", 0.0)
image_width = kwds.pop("image_width", 4.0)
max_iteration = kwds.pop("max_iteration", 500)
pixel_count = kwds.pop("pixel_count", 500)
base_color = kwds.pop("base_color", 'steelblue')
iteration_level = kwds.pop("iteration_level", 1)
number_of_colors = kwds.pop("number_of_colors", 30)
interacts = kwds.pop("interact", False)

if interacts:
return interact.widget(mandelbrot_interact)

base_color = [floor(255 * co) for co in list(Color(base_color).rgb())]
base_color = Color(base_color)

if interact:
from ipywidgets.widgets import FloatSlider, IntSlider, ColorPicker, interact
widgets = dict(
x_center = FloatSlider(min=-1.0, max=1.0, step=EPS,
value=x_center, description="Real center"),
y_center = FloatSlider(min=-1.0, max=1.0, step=EPS,
value=y_center, description="Imag center"),
image_width = FloatSlider(min=EPS, max=4.0, step=EPS,
value=image_width, description="Image width"),
max_iteration = IntSlider(min=0, max=600,
value=max_iteration, description="Iterations"),
pixel_count = IntSlider(min=10, max=600,
value=pixel_count, description="Pixels"),
level_sep = IntSlider(min=1, max=20,
value=iteration_level, description="Color sep"),
color_num = IntSlider(min=1, max=100,
value=number_of_colors, description="# Colors"),
base_color = ColorPicker(value=base_color.html_color(),
description="Base color"),
)
return interact(**widgets).widget(fast_mandelbrot_plot)

return fast_mandelbrot_plot(x_center, y_center, image_width,
max_iteration,
pixel_count, iteration_level,
number_of_colors, base_color)


def mandelbrot_interact(real_center=('Real center:', (-1.0, 1.0, EPS)),
im_center=('Imaginary center:', (-1.0, 1.0, EPS)),
width=('Width of Image:', (EPS, 4.0, EPS)),
iterations=('Max Number of Iterations:', (0, 600)),
pixel_count=("Pixels:", (10, 600)),
level_sep=('Iterations between Colors:', (1, 10)),
color_num=('Number of Colors:', (1, 40)),
image_color=('Base Color:', Color('steelblue'))):
"""
Return the interactive Mandelbrot widget.
Not to be used directly. Use :func:`mandelbrot_plot` instead.
EXAMPLES::
sage: from sage.dynamics.complex_dynamics.mandel_julia import mandelbrot_interact
sage: mandelbrot_plot(interact=True) # indirect doctest
Interactive function ... with 8 widgets
...
"""
return fast_mandelbrot_plot(real_center, im_center, width,
iterations, pixel_count, level_sep,
color_num, image_color)


def external_ray(theta, **kwds):
r"""
Draws the external ray(s) of a given angle (or list of angles)
Expand Down Expand Up @@ -236,12 +226,12 @@ def external_ray(theta, **kwds):
::
sage: external_ray([0, 0.2, 0.4, 0.7]) # long time
sage: external_ray([0, 0.2, 0.4, 0.7])
500x500px 24-bit RGB image
::
sage: external_ray([i/5 for i in range(1,5)]) # long time
sage: external_ray([i/5 for i in range(1,5)])
500x500px 24-bit RGB image
WARNING:
Expand All @@ -250,14 +240,14 @@ def external_ray(theta, **kwds):
which parameters to use when drawing the external ray.
For example, the following is incorrect::
sage: M = mandelbrot_plot(x_center=0) # not tested
sage: external_ray(5/7, image=M) # not tested
sage: M = mandelbrot_plot(x_center=0) # not tested
sage: external_ray(5/7, image=M) # not tested
500x500px 24-bit RGB image
To get the correct external ray, we adjust our parameters::
sage: M = mandelbrot_plot(x_center=0) # not tested
sage: external_ray(5/7, x_center=0, image=M) # not tested
sage: M = mandelbrot_plot(x_center=0)
sage: external_ray(5/7, x_center=0, image=M)
500x500px 24-bit RGB image
.. TODO::
Expand All @@ -273,11 +263,15 @@ def external_ray(theta, **kwds):
sharpness = kwds.get("S", 10)
radial_parameter = kwds.get("R", 100)
precision = kwds.get("prec", 300)
precision = max(precision, -logb(pixel_width * 0.001, 2).round() + 10)
ray_color = kwds.get("ray_color", [255] * 3)
image = kwds.get("image", None)
if image is None:
image = mandelbrot_plot(**kwds)
image = mandelbrot_plot(x_center=x_0,
y_center=y_0,
image_width=plot_width,
pixel_count=pixel_width)

precision = max(precision, -logb(pixel_width * 0.001, 2).round() + 10)

# Make a copy of the bitmap image.
# M = copy(image)
Expand Down Expand Up @@ -328,7 +322,19 @@ def external_ray(theta, **kwds):
return M


def julia_plot(c=-1, **kwds):
def julia_plot(c=-1,
x_center=0.0,
y_center=0.0,
image_width=4.0,
max_iteration=500,
pixel_count=500,
base_color='steelblue',
iteration_level=1,
number_of_colors=50,
point_color='yellow',
interact=False,
mandelbrot=True,
period=None):
r"""
Plots the Julia set of a given complex `c` value. Users can specify whether
they would like to display the Mandelbrot side by side with the Julia set.
Expand All @@ -354,8 +360,6 @@ def julia_plot(c=-1, **kwds):
- ``c`` -- complex (optional - default: ``-1``), complex point `c` that
determines the Julia set.
kwds:
- ``period`` -- list (optional - default: ``None``), returns the Julia set
for a random `c` value with the given (formal) cycle structure.
Expand Down Expand Up @@ -411,7 +415,7 @@ def julia_plot(c=-1, **kwds):
set ``interact`` to ``True``::
sage: julia_plot(interact=True)
Interactive function ... with 12 widgets...
interactive(children=(FloatSlider(value=-1.0, description=u'Real c'...
To return the Julia set of a random `c` value with (formal) cycle structure
`(2,3)`, set ``period = [2,3]``::
Expand All @@ -431,19 +435,6 @@ def julia_plot(c=-1, **kwds):
....: for c in c_values:
....: julia_plot(c)
"""
x_center = kwds.pop("x_center", 0.0)
y_center = kwds.pop("y_center", 0.0)
image_width = kwds.pop("image_width", 4.0)
max_iteration = kwds.pop("max_iteration", 500)
pixel_count = kwds.pop("pixel_count", 500)
base_color = kwds.pop("base_color", 'steelblue')
iteration_level = kwds.pop("iteration_level", 1)
number_of_colors = kwds.pop("number_of_colors", 50)
point_color = kwds.pop("point_color", 'tomato')
interacts = kwds.pop("interact", False)
mandelbrot = kwds.pop("mandelbrot", True)
period = kwds.pop("period", None)

if period is not None:
R = PolynomialRing(QQ, 'c')
c = R.gen()
Expand All @@ -452,14 +443,44 @@ def julia_plot(c=-1, **kwds):
L = f.dynatomic_polynomial(period).subs({x: 0, y: 1}).roots(ring=CC)
c = L[randint(0, len(L) - 1)][0]

c_real = CC(c).real()
c_imag = CC(c).imag()

if interacts:
return interact.widget(julia_interact)
c = CC(c)
c_real = c.real()
c_imag = c.imag()

base_color = Color(base_color)
point_color = Color(point_color)

if interact:
from ipywidgets.widgets import FloatSlider, IntSlider, ColorPicker, interact
widgets = dict(
c_real = FloatSlider(min=-2.0, max=2.0, step=EPS,
value=c_real, description="Real c"),
c_imag = FloatSlider(min=-2.0, max=2.0, step=EPS,
value=c_imag, description="Imag c"),
x_center = FloatSlider(min=-1.0, max=1.0, step=EPS,
value=x_center, description="Real center"),
y_center = FloatSlider(min=-1.0, max=1.0, step=EPS,
value=y_center, description="Imag center"),
image_width = FloatSlider(min=EPS, max=4.0, step=EPS,
value=image_width, description="Image width"),
max_iteration = IntSlider(min=0, max=600,
value=max_iteration, description="Iterations"),
pixel_count = IntSlider(min=10, max=600,
value=pixel_count, description="Pixels"),
level_sep = IntSlider(min=1, max=20,
value=iteration_level, description="Color sep"),
color_num = IntSlider(min=1, max=100,
value=number_of_colors, description="# Colors"),
base_color = ColorPicker(value=base_color.html_color(),
description="Base color"),
)
if mandelbrot:
widgets["point_color"] = ColorPicker(value=point_color.html_color(),
description="Point color")
return interact(**widgets).widget(julia_helper)
else:
return interact(**widgets).widget(fast_julia_plot)

base_color = [floor(255 * co) for co in list(Color(base_color).rgb())]
point_color = [floor(255 * co) for co in list(Color(point_color).rgb())]
if mandelbrot:
return julia_helper(c_real, c_imag, x_center, y_center,
image_width, max_iteration, pixel_count,
Expand All @@ -471,44 +492,3 @@ def julia_plot(c=-1, **kwds):
image_width, max_iteration, pixel_count,
iteration_level,
number_of_colors, base_color)


def julia_interact(cx=('Re(c):', (-1.0, 1.0, EPS)),
cy=('Im(c):', (-1.0, 1.0, EPS)),
real_center=('Real Center:', (-1.0, 1.0, EPS)),
im_center=('Imaginary Center:', (-1.0, 1.0, EPS)),
width=('Width of Image:', (EPS, 4.0, EPS)),
iterations=('Max Number of Iterations:', (1, 600)),
pixel_count=("Pixels:", (1, 600)),
level_sep=('Iterations between Colors', (1, 10)),
color_num=('Number of Colors:', (1, 40)),
image_color=("Image Color:", Color('tomato')),
pt_color=("Point Color:", Color('steelblue')),
mandel=('Mandelbrot set:', False)):
"""
Return the interactive Julia widget.
Not to be used directly. Use :func:`julia_plot` instead.
EXAMPLES::
sage: from sage.dynamics.complex_dynamics.mandel_julia import julia_interact
sage: julia_plot(interact=True) # indirect doctest
Interactive function ... with 12 widgets
...
"""
if mandel:
return julia_helper(cx, cy,
real_center, im_center,
width, iterations,
pixel_count, level_sep,
color_num,
image_color,
pt_color)
else:
return fast_julia_plot(cx, cy,
real_center, im_center,
width, iterations,
pixel_count, level_sep,
color_num,
image_color)

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