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.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
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.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_api_gallery_topology_data_plot_02_cech_nerve_theorem.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_data_plot_02_cech_nerve_theorem.py:


Čech complexes and the nerve theorem (1932-1948)
================================================

Eduard Čech built homology from the *nerve* of a cover: a simplex for
every set of cover sets that share a point. Borsuk's nerve theorem
(1948) says that when every intersection is contractible, as for round
discs, the nerve has the homotopy type of the union. Below, the Čech
complex of discs around noisy samples of two touching rings has the
same Betti numbers as the union of the discs, counted independently from
a pixel image. The Vietoris-Rips complex at the same radius also fills
in triangles whose three discs meet pairwise but share no common point.
At :math:`r = 0.6` that closes the small ring's hole while the discs
still leave it open.

.. GENERATED FROM PYTHON SOURCE LINES 18-51



.. image-sg:: /api/gallery/topology/data/images/sphx_glr_plot_02_cech_nerve_theorem_001.png
   :alt: Nerve theorem: the Čech complex has the homotopy type of the union of discs, r = 0.12: discs (9, 0), Čech (9, 0), r = 0.3: discs (1, 2), Čech (1, 2), r = 0.6: discs (1, 2), Čech (1, 2), Rips at the same r: (9, 0), Rips at the same r: (1, 2), Rips at the same r: (1, 1)
   :srcset: /api/gallery/topology/data/images/sphx_glr_plot_02_cech_nerve_theorem_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    r = 0.12: union of discs (9, 0), Čech (9, 0), Rips (9, 0)
    r = 0.3: union of discs (1, 2), Čech (1, 2), Rips (1, 2)
    r = 0.6: union of discs (1, 2), Čech (1, 2), Rips (1, 1)

    Text(0.5, 0.98, 'Nerve theorem: the Čech complex has the homotopy type of the union of discs')





|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.topology import betti_numbers, cech_complex, planar_betti_numbers, vietoris_rips_complex
    from mathematicskit.topology.visualizers import plot_complex

    rng = np.random.default_rng(1)
    t, s = np.sort(rng.uniform(0, 2 * np.pi, 60)), np.sort(rng.uniform(0, 2 * np.pi, 40))
    points = np.vstack([np.column_stack([np.cos(t), np.sin(t)]), np.column_stack([1.6 + 0.7 * np.cos(s), 0.7 * np.sin(s)])])
    points += rng.normal(0, 0.03, points.shape)

    xs = np.linspace(-1.6, 2.8, 700)
    ys = np.linspace(-1.6, 1.6, 520)
    gx, gy = np.meshgrid(xs, ys)
    fig, axes = plt.subplots(2, 3, figsize=(14, 8))
    for col, r in enumerate((0.12, 0.3, 0.6)):
        union = np.zeros_like(gx, dtype=bool)
        for p in points:
            union |= np.hypot(gx - p[0], gy - p[1]) <= r
        C, R = cech_complex(points, r), vietoris_rips_complex(points, r)
        b_union, b_cech, b_rips = planar_betti_numbers(union), betti_numbers(C, field=2)[:2], betti_numbers(R, field=2)[:2]
        print(f"r = {r}: union of discs {b_union}, Čech {b_cech}, Rips {b_rips}")
        ax = axes[0, col]
        ax.contourf(gx, gy, union, levels=[0.5, 1.5], colors=["tab:blue"], alpha=0.3)
        plot_complex(C, ax=ax, alpha=0.4, vertex_size=4)
        ax.set_title(f"r = {r}: discs {b_union}, Čech {b_cech}")
        plot_complex(R, ax=axes[1, col], face_color="tab:orange", alpha=0.4, vertex_size=4)
        axes[1, col].set_title(f"Rips at the same r: {b_rips}")
    for ax in axes.ravel():
        ax.set_xlim(xs[0], xs[-1])
        ax.set_ylim(ys[0], ys[-1])
        ax.axis("off")
    fig.suptitle("Nerve theorem: the Čech complex has the homotopy type of the union of discs")


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

   **Total running time of the script:** (0 minutes 0.893 seconds)


.. _sphx_glr_download_api_gallery_topology_data_plot_02_cech_nerve_theorem.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/data/plot_02_cech_nerve_theorem.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_cech_nerve_theorem.ipynb <plot_02_cech_nerve_theorem.ipynb>`

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      :download:`Download Python source code: plot_02_cech_nerve_theorem.py <plot_02_cech_nerve_theorem.py>`

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

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