.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/surfaces/plot_04_morse_banchoff_critical_points.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_topology_surfaces_plot_04_morse_banchoff_critical_points.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_topology_surfaces_plot_04_morse_banchoff_critical_points.py:


Morse theory and Banchoff's critical points (1925-1967)
=======================================================

Morse related the critical points of a function on a manifold to its
topology: the alternating count of minima, saddles and maxima is the
Euler characteristic. Banchoff found the polyhedral version. Charge
each simplex to its highest vertex, and a vertex's index is the
alternating count of what it receives. On an upright torus the height
has one minimum, two saddles and one maximum (:math:`1 - 2 + 1 = 0`).
A wobblier height function has more critical points, but the
alternating sum stays at 0.

.. GENERATED FROM PYTHON SOURCE LINES 16-43



.. image-sg:: /api/gallery/topology/surfaces/images/sphx_glr_plot_04_morse_banchoff_critical_points_001.png
   :alt: Critical points of any height function sum to χ(torus) = 0, height x: 1 - 2 + 1 = 0, wobbly height: 4 - 8 + 4 = 0
   :srcset: /api/gallery/topology/surfaces/images/sphx_glr_plot_04_morse_banchoff_critical_points_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    height x: 1 minima, 2 saddles, 1 maxima; total index 0
    wobbly height: 4 minima, 8 saddles, 4 maxima; total index 0

    Text(0.5, 0.98, 'Critical points of any height function sum to χ(torus) = 0')





|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.topology import critical_points, torus
    from mathematicskit.topology.visualizers import plot_complex

    T = torus(32, 16)
    x, y, z = T.coordinates.T
    heights = {
        "height x": x + 1e-3 * z,
        "wobbly height": x + 0.6 * np.sin(3 * y) + 0.3 * z,
    }

    fig = plt.figure(figsize=(12, 5))
    for i, (name, h) in enumerate(heights.items()):
        result = critical_points(T, h)
        print(f"{name}: {len(result.minima)} minima, {len(result.saddles)} saddles, {len(result.maxima)} maxima; total index {result.euler_characteristic}")
        ax = fig.add_subplot(1, 2, i + 1, projection="3d", computed_zorder=False)
        plot_complex(T, ax=ax, face_values=[h[list(t)].mean() for t in T.simplices(2)], alpha=0.55, vertex_size=0)
        for vertices, color, label in ((result.minima, "tab:blue", "minimum"), (result.saddles, "tab:red", "saddle"), (result.maxima, "gold", "maximum")):
            ax.scatter(*T.coordinates[vertices].T, s=60, color=color, edgecolors="k", depthshade=False, label=label)
        ax.view_init(elev=40, azim=-90)
        ax.set_box_aspect((1, 1, 1), zoom=1.4)
        ax.set_axis_off()
        ax.set_title(f"{name}: {len(result.minima)} - {len(result.saddles)} + {len(result.maxima)} = {result.euler_characteristic}")
        ax.legend(loc="lower left", fontsize=8)
    fig.suptitle("Critical points of any height function sum to χ(torus) = 0")


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

   **Total running time of the script:** (0 minutes 0.080 seconds)


.. _sphx_glr_download_api_gallery_topology_surfaces_plot_04_morse_banchoff_critical_points.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/topology/surfaces/plot_04_morse_banchoff_critical_points.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_04_morse_banchoff_critical_points.ipynb <plot_04_morse_banchoff_critical_points.ipynb>`

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

      :download:`Download Python source code: plot_04_morse_banchoff_critical_points.py <plot_04_morse_banchoff_critical_points.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_04_morse_banchoff_critical_points.zip <plot_04_morse_banchoff_critical_points.zip>`


.. only:: html

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

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