def julia_plot(f=None, **kwds):
r"""
Plots the Julia set of a given polynomial ``f``. Users can specify whether
they would like to display the Mandelbrot side by side with the Julia set
with the ``mandelbrot`` argument. If ``f`` is not specified, this method
defaults to `f(z) = z^2-1`.
The Julia set of a polynomial ``f`` is the set of complex numbers `z` for
which the function `f(z)` is bounded under iteration. The Julia set can
be visualized by plotting each point in the set in the complex plane.
Julia sets are examples of fractals when plotted in the complex plane.
ALGORITHM:
Let `R_c = \bigl(1 + \sqrt{1 + 4|c|}\bigr)/2` if the polynomial is of the
form `f(z) = z^2 + c`; otherwise, let `R_c = 2`.
For every `p \in \mathbb{C}`, if `|f^{k}(p)| > R_c` for some `k \geq 0`,
then `f^{n}(p) \to \infty`. Let `N` be the maximum number of iterations.
Compute the first `N` points on the orbit of `p` under `f`. If for
any `k < N`, `|f^{k}(p)| > R_c`, we stop the iteration and assign a color
to the point `p` based on how quickly `p` escaped to infinity under
iteration of `f`. If `|f^{i}(p)| \leq R_c` for all `i \leq N`, we assume
`p` is in the Julia set and assign the point `p` the color black.
INPUT:
- ``f`` -- input polynomial (optional - default: ``z^2 - 1``).
- ``period`` -- list (optional - default: ``None``), returns the Julia set
for a random `c` value with the given (formal) cycle structure.
- ``mandelbrot`` -- boolean (optional - default: ``True``), when set to
``True``, an image of the Mandelbrot set is appended to the right of the
Julia set.
- ``point_color`` -- RGB color (optional - default: ``'tomato'``),
color of the point `c` in the Mandelbrot set (any valid input for Color).
- ``x_center`` -- double (optional - default: ``-1.0``), 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 `f(z)`.
- ``pixel_count`` -- long (optional - default: ``500``), side length of
image in number of pixels.
- ``base_color`` -- hex color (optional - default: ``'steelblue'``), color
used to determine the coloring of set (any valid input for Color).
- ``level_sep`` -- long (optional - default: 1), number of iterations
between each color level.
- ``number_of_colors`` -- long (optional - default: 30), number of colors
used to plot image.
- ``interact`` -- boolean (optional - default: ``False``), controls whether
plot will have interactive functionality.
OUTPUT:
24-bit RGB image of the Julia set in the complex plane.
.. TODO::
Implement the side-by-side Mandelbrot-Julia plots for general one-parameter families
of polynomials.
EXAMPLES:
The default ``f`` is `z^2 - 1`::
sage: julia_plot()
1001x500px 24-bit RGB image
To display only the Julia set, set ``mandelbrot`` to ``False``::
sage: julia_plot(mandelbrot=False)
500x500px 24-bit RGB image
::
sage: R.<z> = CC[]
sage: f = z^3 - z + 1
sage: julia_plot(f)
500x500px 24-bit RGB image
To display an interactive plot of the Julia set in the Notebook,
set ``interact`` to ``True``. (This is only implemented for polynomials of
the form ``f = z^2 + c``)::
sage: julia_plot(interact=True)
interactive(children=(FloatSlider(value=-1.0, description=u'Real c'...
::
sage: R.<z> = CC[]
sage: f = z^2 + 1/2
sage: julia_plot(f,interact=True)
interactive(children=(FloatSlider(value=0.5, description=u'Real c'...
To return the Julia set of a random `c` value with (formal) cycle structure
`(2,3)`, set ``period = [2,3]``::
sage: julia_plot(period=[2,3])
1001x500px 24-bit RGB image
To return all of the Julia sets of `c` values with (formal) cycle structure
`(2,3)`::
sage: period = [2,3] # not tested
....: R.<c> = QQ[]
....: P.<x,y> = ProjectiveSpace(R,1)
....: f = DynamicalSystem([x^2+c*y^2, y^2])
....: L = f.dynatomic_polynomial(period).subs({x:0,y:1}).roots(ring=CC)
....: c_values = [k[0] for k in L]
....: for c in c_values:
....: julia_plot(c)
Polynomial maps can be defined over a polynomial ring or a fraction field,
so long as ``f`` is polynomial::
sage: R.<z> = CC[]
sage: f = z^2 - 1
sage: julia_plot(f)
1001x500px 24-bit RGB image
::
sage: R.<z> = CC[]
sage: K = R.fraction_field(); z = K.gen()
sage: f = z^2-1
sage: julia_plot(f)
1001x500px 24-bit RGB image
Interact functionality is not implemented if the polynomial is not of the
form `f = z^2 + c`::
sage: R.<z> = CC[]
sage: f = z^3 + 1
sage: julia_plot(f, interact=True)
Traceback (most recent call last):
...
NotImplementedError: The interactive plot is only implemented for ...
"""
# extract keyword arguments
period = kwds.pop("period", None)
mandelbrot = kwds.pop("mandelbrot", True)
point_color = kwds.pop("point_color", 'tomato')
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')
level_sep = kwds.pop("level_sep", 1)
number_of_colors = kwds.pop("number_of_colors", 30)
interacts = kwds.pop("interact", False)
f_is_default_after_all = None
if period: # pick a random c with the specified period
R = PolynomialRing(CC, 'c')
c = R.gen()
x, y = ProjectiveSpace(R, 1, 'x,y').gens()
F = DynamicalSystem([x**2 + c * y**2, y**2])
L = F.dynatomic_polynomial(period).subs({x: 0, y: 1}).roots(ring=CC)
c = L[randint(0, len(L) - 1)][0]
base_color = Color(base_color)
point_color = Color(point_color)
EPS = 0.00001
if f is not None and period is None: # f user-specified and no period given
# try to coerce f to live in a polynomial ring
S = PolynomialRing(CC, names='z')
z = S.gen()
try:
f_poly = S(f)
except TypeError:
R = f.parent()
if not (R.is_integral_domain() and
(CC.is_subring(R) or CDF.is_subring(R))):
raise ValueError('Given `f` must be a complex polynomial.')
else:
raise NotImplementedError(
'Julia sets not implemented for rational functions.')
if (f_poly - z * z) in CC: # f is specified and of the form z^2 + c.
f_is_default_after_all = True
c = f_poly - z * z
else: # f is specified and not of the form z^2 + c
if interacts:
raise NotImplementedError(
"The interactive plot is only implemented for "
"polynomials of the form f = z^2 + c.")
else:
return general_julia(f_poly, x_center, y_center, image_width,
max_iteration, pixel_count, level_sep,
number_of_colors, base_color)
# otherwise we can use fast_julia_plot for z^2 + c
if f_is_default_after_all or f is None or period is not None:
# specify default c = -1 value if f and period were not specified
if not f_is_default_after_all and period is None:
c = -1
c = CC(c)
c_real = c.real()
c_imag = c.imag()
if interacts: # set widgets
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="Width"),
max_iteration=IntSlider(min=0,
max=1000,
value=max_iteration,
description="Iterations"),
pixel_count=IntSlider(min=10,
max=1000,
value=pixel_count,
description="Pixels"),
level_sep=IntSlider(min=1,
max=20,
value=level_sep,
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)
elif mandelbrot: # non-interactive with mandelbrot
return julia_helper(c_real, c_imag, x_center, y_center,
image_width, max_iteration, pixel_count,
level_sep, number_of_colors, base_color,
point_color)
else: # non-interactive without mandelbrot
return fast_julia_plot(c_real, c_imag, x_center, y_center,
image_width, max_iteration, pixel_count,
level_sep, number_of_colors, base_color)
