fractals.julia_sets
===================

.. py:module:: fractals.julia_sets

.. autoapi-nested-parse::

   Author Alexandre De Zotti

   Draws Julia sets of quadratic polynomials and exponential maps.
    More specifically, this iterates the function a fixed number of times
    then plots whether the absolute value of the last iterate is greater than
    a fixed threshold (named "escape radius"). For the exponential map this is not
    really an escape radius but rather a convenient way to approximate the Julia
    set with bounded orbits.

   The examples presented here are:
   - The Cauliflower Julia set, see e.g.
   https://en.wikipedia.org/wiki/File:Julia_z2%2B0,25.png
   - Other examples from https://en.wikipedia.org/wiki/Julia_set
   - An exponential map Julia set, ambiantly homeomorphic to the examples in
   https://www.math.univ-toulouse.fr/~cheritat/GalII/galery.html
    and
   https://ddd.uab.cat/pub/pubmat/02141493v43n1/02141493v43n1p27.pdf

   Remark: Some overflow runtime warnings are suppressed. This is because of the
    way the iteration loop is implemented, using numpy's efficient computations.
    Overflows and infinites are replaced after each step by a large number.



Attributes
----------

.. autoapisummary::

   fractals.julia_sets.c_cauliflower
   fractals.julia_sets.c_exponential
   fractals.julia_sets.c_polynomial_1
   fractals.julia_sets.c_polynomial_2
   fractals.julia_sets.nb_iterations
   fractals.julia_sets.nb_pixels
   fractals.julia_sets.window_size
   fractals.julia_sets.z_0


Functions
---------

.. autoapisummary::

   fractals.julia_sets.eval_exponential
   fractals.julia_sets.eval_quadratic_polynomial
   fractals.julia_sets.ignore_overflow_warnings
   fractals.julia_sets.iterate_function
   fractals.julia_sets.prepare_grid
   fractals.julia_sets.show_results


Module Contents
---------------

.. py:function:: eval_exponential(c_parameter: complex, z_values: numpy.ndarray) -> numpy.ndarray

   Evaluate $e^z + c$.
   >>> float(eval_exponential(0, 0))
   1.0
   >>> bool(abs(eval_exponential(1, np.pi*1.j)) < 1e-15)
   True
   >>> bool(abs(eval_exponential(1.j, 0)-1-1.j) < 1e-15)
   True


.. py:function:: eval_quadratic_polynomial(c_parameter: complex, z_values: numpy.ndarray) -> numpy.ndarray

   >>> eval_quadratic_polynomial(0, 2)
   4
   >>> eval_quadratic_polynomial(-1, 1)
   0
   >>> round(eval_quadratic_polynomial(1.j, 0).imag)
   1
   >>> round(eval_quadratic_polynomial(1.j, 0).real)
   0


.. py:function:: ignore_overflow_warnings() -> None

   Ignore some overflow and invalid value warnings.

   >>> ignore_overflow_warnings()


.. py:function:: iterate_function(eval_function: collections.abc.Callable[[Any, numpy.ndarray], numpy.ndarray], function_params: Any, nb_iterations: int, z_0: numpy.ndarray, infinity: float | None = None) -> numpy.ndarray

   Iterate the function "eval_function" exactly nb_iterations times.
   The first argument of the function is a parameter which is contained in
   function_params. The variable z_0 is an array that contains the initial
   values to iterate from.
   This function returns the final iterates.

   >>> iterate_function(eval_quadratic_polynomial, 0, 3, np.array([0,1,2])).shape
   (3,)
   >>> complex(np.round(iterate_function(eval_quadratic_polynomial,
   ... 0,
   ... 3,
   ... np.array([0,1,2]))[0]))
   0j
   >>> complex(np.round(iterate_function(eval_quadratic_polynomial,
   ... 0,
   ... 3,
   ... np.array([0,1,2]))[1]))
   (1+0j)
   >>> complex(np.round(iterate_function(eval_quadratic_polynomial,
   ... 0,
   ... 3,
   ... np.array([0,1,2]))[2]))
   (256+0j)


.. py:function:: prepare_grid(window_size: float, nb_pixels: int) -> numpy.ndarray

   Create a grid of complex values of size nb_pixels*nb_pixels with real and
    imaginary parts ranging from -window_size to window_size (inclusive).
   Returns a numpy array.

   >>> prepare_grid(1,3)
   array([[-1.-1.j, -1.+0.j, -1.+1.j],
          [ 0.-1.j,  0.+0.j,  0.+1.j],
          [ 1.-1.j,  1.+0.j,  1.+1.j]])


.. py:function:: show_results(function_label: str, function_params: Any, escape_radius: float, z_final: numpy.ndarray) -> None

   Plots of whether the absolute value of z_final is greater than
   the value of escape_radius. Adds the function_label and function_params to
   the title.

   >>> show_results('80', 0, 1, np.array([[0,1,.5],[.4,2,1.1],[.2,1,1.3]]))


.. py:data:: c_cauliflower
   :value: (0.25+0j)


.. py:data:: c_exponential
   :value: -2.0


.. py:data:: c_polynomial_1
   :value: (-0.4+0.6j)


.. py:data:: c_polynomial_2
   :value: (-0.1+0.651j)


.. py:data:: nb_iterations
   :value: 56


.. py:data:: nb_pixels
   :value: 666


.. py:data:: window_size
   :value: 2.0


.. py:data:: z_0