.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "api/gallery/chaos/billiards/plot_ellipse_billiard.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code. .. rst-class:: sphx-glr-example-title .. _sphx_glr_api_gallery_chaos_billiards_plot_ellipse_billiard.py: Ellipse Billiard (Integrable, with Caustics) =============================================== A billiard particle moves in a straight line at constant speed inside a closed boundary and undergoes specular reflection, :math:`\mathbf{v}' = \mathbf{v} - 2(\mathbf{v}\cdot\mathbf{n})\,\mathbf{n}`, whenever it strikes the boundary. Here the boundary is the ellipse .. math:: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1, with semi-major axis :math:`a` and semi-minor axis :math:`b`, and foci at :math:`(\pm c, 0)`, :math:`c = \sqrt{a^2 - b^2}`. The elliptical billiard is integrable, like the Circle and Rectangle, but illustrates a richer piece of the theory: every trajectory stays tangent to a single *confocal caustic* for all time -- either a confocal ellipse (for trajectories that never pass between the two foci) or a confocal hyperbola (for trajectories that do). This example plots a trajectory of each kind and the resulting Poincare section. .. GENERATED FROM PYTHON SOURCE LINES 23-33 .. code-block:: Python import matplotlib.pyplot as plt from physicskit.chaos.systems.billiards import EllipseBilliard from physicskit.chaos.visualizers.dynamic_plots import animate_billiard_trajectory from physicskit.chaos.visualizers.phase_space import plot_billiard_trajectory, plot_poincare_section billiard = EllipseBilliard(semi_major=1.5, semi_minor=1.0) f1, f2 = billiard.foci() .. GENERATED FROM PYTHON SOURCE LINES 34-38 Animation --------- Watch the ray trace out its confocal elliptical caustic -- the envelope of lines it stays perpetually tangent to -- one bounce at a time. .. GENERATED FROM PYTHON SOURCE LINES 38-42 .. code-block:: Python anim = animate_billiard_trajectory(billiard, pos=(0.0, 0.0), vel=(1.0, 0.15), n_bounces=100, interval=50) plt.show() .. container:: sphx-glr-animation .. raw:: html

.. GENERATED FROM PYTHON SOURCE LINES 43-47 To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:: anim.save("ellipse_billiard_animation.gif", writer="pillow", fps=20) .. GENERATED FROM PYTHON SOURCE LINES 49-54 Two families of caustics --------------------------- A trajectory that never crosses the segment joining the foci stays tangent to a confocal *ellipse* (left); one that does cross it stays tangent to a confocal *hyperbola* (right). .. GENERATED FROM PYTHON SOURCE LINES 54-65 .. code-block:: Python fig, axes = plt.subplots(1, 2, figsize=(12, 6)) plot_billiard_trajectory(billiard, pos=(0.0, 0.0), vel=(1.0, 0.15), n_bounces=100, ax=axes[0]) axes[0].plot(*f1, "k.", *f2, "k.") axes[0].set_title("Elliptical caustic") plot_billiard_trajectory(billiard, pos=(0.0, 0.0), vel=(0.15, 1.0), n_bounces=100, ax=axes[1]) axes[1].plot(*f1, "k.", *f2, "k.") axes[1].set_title("Hyperbolic caustic") fig.tight_layout() .. image-sg:: /api/gallery/chaos/billiards/images/sphx_glr_plot_ellipse_billiard_002.png :alt: Elliptical caustic, Hyperbolic caustic :srcset: /api/gallery/chaos/billiards/images/sphx_glr_plot_ellipse_billiard_002.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 66-72 Poincare section ---------------- Every ray traces out a smooth invariant curve, just as for the Circle and Rectangle billiards -- the signature of integrability. Rays confined to elliptical caustics produce closed curves; rays confined to hyperbolic caustics produce the "wavy" curves crossing the ``sin(phi) = 0`` axis. .. GENERATED FROM PYTHON SOURCE LINES 72-75 .. code-block:: Python fig2, ax2 = plot_poincare_section(billiard, n_rays=15, n_bounces=200) plt.show() .. image-sg:: /api/gallery/chaos/billiards/images/sphx_glr_plot_ellipse_billiard_003.png :alt: EllipseBilliard Poincare section :srcset: /api/gallery/chaos/billiards/images/sphx_glr_plot_ellipse_billiard_003.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 8.846 seconds) .. _sphx_glr_download_api_gallery_chaos_billiards_plot_ellipse_billiard.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: plot_ellipse_billiard.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: plot_ellipse_billiard.py ` .. container:: sphx-glr-download sphx-glr-download-zip :download:`Download zipped: plot_ellipse_billiard.zip ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_