.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "matplotlib/plot_circular_billard.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        Click :ref:`here <sphx_glr_download_matplotlib_plot_circular_billard.py>`
        to download the full example code

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_matplotlib_plot_circular_billard.py:


(classical) circular billard
===============================

.. GENERATED FROM PYTHON SOURCE LINES 5-83



.. image-sg:: /matplotlib/images/sphx_glr_plot_circular_billard_001.png
   :alt: plot circular billard
   :srcset: /matplotlib/images/sphx_glr_plot_circular_billard_001.png
   :class: sphx-glr-single-img





.. code-block:: default



    import numpy as np
    import matplotlib.pyplot as plt

    fig, ax = plt.subplots(1,1, figsize=(6,6))

    theta = np.linspace(-np.pi, np.pi, 100)
    z = np.exp(1j*theta)
    ax.plot(z.real, z.imag, '-')
    ax.set_xlim(-1,1)
    ax.set_ylim(-1,1)

    def df(x):
        return np.array([2*x[0], 2*x[1]])

    def intersection(vec0, coord):
        a = vec0[1]/vec0[0]
        if np.abs(a) > 1e7:
            z = np.array([x1, -y0])

        x0 = coord[0]
        y0 = coord[1]
        sg = np.sign(vec0[0])
        D = 1 - a**2*(-1+x0**2)+ 2*a*x0*y0 -y0**2 
        x1 = - ( -a**2*x0 + a*y0 - sg * np.sqrt(D) ) / (1 + a**2)

        y1 = (y0 -a * ( x0 - sg* np.sqrt(D) ) )/ (1 + a**2)    
        z = np.array([x1, y1])
        return z

    #x0 = np.array([0, 1/np.sqrt(2)])
    x0 = np.array([0.5, 0.1])
    vec0 = np.array([-0.2, 0.2])

    ini_x0 = x0.copy()
    ini_vec0 = vec0.copy()

    x1 = intersection(vec0, x0)

    traj = [np.array([]), np.array([])]

    traj[0] = np.append(traj[0], x0[0])
    traj[1] = np.append(traj[1], x0[1])

    traj[0] = np.append(traj[0], x1[0])
    traj[1] = np.append(traj[1], x1[1])

    n = -df(x1)
    pn = -np.dot(vec0, n)/np.dot(n, n) * n
    vec1 = vec0 + 2*pn 

    #ax.quiver(x1[0], x1[1], n[0], n[1], angles='xy', scale_units='xy', scale=10)
    #ax.quiver(x1[0], x1[1], vec1[0], vec1[1], angles='xy', scale_units='xy', scale=1)

    for n in range(30):
        x0 = x1.copy()
        vec0 = vec1.copy()

        x1 = intersection(vec0, x0)
        traj[0] = np.append(traj[0], x1[0])
        traj[1] = np.append(traj[1], x1[1])    

        n = -df(x1)
        pn = -np.dot(vec0, n)/np.dot(n, n) * n
        vec1 = vec0 + 2*pn 

        xx = np.linspace(x0[0], x1[0], 2)
        yy = vec0[1]/vec0[0]*(xx - x0[0]) + x0[1]


    ax.plot(traj[0], traj[1],'-', alpha=0.7)

    ax.plot(ini_x0[0], ini_x0[1], 'ok')
    ax.quiver(ini_x0[0], ini_x0[1], ini_vec0[0], ini_vec0[1], angles='xy', scale_units='xy', scale=0.5)


    ax.grid()
    plt.show()

.. rst-class:: sphx-glr-timing

   **Total running time of the script:** ( 0 minutes  0.098 seconds)


.. _sphx_glr_download_matplotlib_plot_circular_billard.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example


    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_circular_billard.py <plot_circular_billard.py>`

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_circular_billard.ipynb <plot_circular_billard.ipynb>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_