.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/data/plot_01_vietoris_rips_complex.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_data_plot_01_vietoris_rips_complex.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_01_vietoris_rips_complex.py:


Vietoris's complex of a point cloud (1927)
==========================================

To define homology for any compact metric space, Leopold Vietoris
joined every set of points that are pairwise close. Today this is the
Vietoris-Rips complex: at scale :math:`r` (balls of radius :math:`r`),
a simplex for every set of points at most :math:`2r` apart. Sampled from
a figure eight, the complex at small scales is dust. At the right
scale it has :math:`\beta = (1, 2)`, the figure eight's two loops. At
large scales everything fills in. Triangles are enough to count loops,
and mod-2 coefficients keep the rank computation fast.

.. GENERATED FROM PYTHON SOURCE LINES 16-36



.. image-sg:: /api/gallery/topology/data/images/sphx_glr_plot_01_vietoris_rips_complex_001.png
   :alt: Vietoris-Rips complexes of a noisy figure eight, r = 0.05: $(\beta_0, \beta_1)$ = (46, 0), r = 0.15: $(\beta_0, \beta_1)$ = (13, 0), r = 0.3: $(\beta_0, \beta_1)$ = (1, 1), r = 0.6: $(\beta_0, \beta_1)$ = (1, 2)
   :srcset: /api/gallery/topology/data/images/sphx_glr_plot_01_vietoris_rips_complex_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    r = 0.05: f = (80, 46, 14), (beta_0, beta_1) = (46, 0)
    r = 0.15: f = (80, 164, 170), (beta_0, beta_1) = (13, 0)
    r = 0.30: f = (80, 362, 819), (beta_0, beta_1) = (1, 1)
    r = 0.60: f = (80, 828, 4453), (beta_0, beta_1) = (1, 2)

    Text(0.5, 0.98, 'Vietoris-Rips complexes of a noisy figure eight')





|

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

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

    rng = np.random.default_rng(4)
    t = rng.uniform(0, 2 * np.pi, 80)
    points = np.column_stack([np.sin(t), np.sin(t) * np.cos(t)]) * [2, 2] + rng.normal(0, 0.04, (80, 2))

    radii = [0.05, 0.15, 0.3, 0.6]
    fig, axes = plt.subplots(1, 4, figsize=(15, 4))
    for ax, r in zip(axes, radii, strict=True):
        K = vietoris_rips_complex(points, r, max_dim=2)
        betti = betti_numbers(K, field=2)[:2]
        print(f"r = {r:4.2f}: f = {K.f_vector}, (beta_0, beta_1) = {betti}")
        plot_complex(K, ax=ax, alpha=0.25, vertex_size=6)
        ax.set_title(f"r = {r}: $(\\beta_0, \\beta_1)$ = {betti}")
        ax.axis("off")
    fig.suptitle("Vietoris-Rips complexes of a noisy figure eight")


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

   **Total running time of the script:** (0 minutes 0.104 seconds)


.. _sphx_glr_download_api_gallery_topology_data_plot_01_vietoris_rips_complex.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_01_vietoris_rips_complex.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_vietoris_rips_complex.ipynb <plot_01_vietoris_rips_complex.ipynb>`

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

      :download:`Download Python source code: plot_01_vietoris_rips_complex.py <plot_01_vietoris_rips_complex.py>`

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

      :download:`Download zipped: plot_01_vietoris_rips_complex.zip <plot_01_vietoris_rips_complex.zip>`


.. only:: html

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

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