.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/topology/curves/plot_01_gauss_linking_number.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_curves_plot_01_gauss_linking_number.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_curves_plot_01_gauss_linking_number.py:


Gauss's linking number (1833)
=============================

Gauss's double integral
:math:`\frac{1}{4\pi}\oint\oint \frac{(r_1 - r_2)\cdot(dr_1 \times dr_2)}{|r_1 - r_2|^3}`
counts how many times one closed curve winds around another. It is an
integer that no deformation can change without passing one curve
through the other. Below: the unlink (0), the Hopf link (:math:`\pm 1`,
the sign set by the orientations), and torus links whose components
wind :math:`n` times around each other.

.. GENERATED FROM PYTHON SOURCE LINES 15-40

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.topology import linking_number

    t = np.linspace(0, 2 * np.pi, 300, endpoint=False)
    ring = np.column_stack([np.cos(t), np.sin(t), 0 * t])
    links = {
        "unlink": (ring, np.column_stack([3 + np.cos(t), 0 * t, np.sin(t)])),
        "Hopf link": (ring, np.column_stack([1 + np.cos(t), 0 * t, np.sin(t)])),
        "Hopf link, reversed": (ring, np.column_stack([1 + np.cos(t), 0 * t, np.sin(t)])[::-1]),
    }


    def torus_link(n, phase):
        v = n * t + phase
        return np.column_stack([(2 + np.cos(v)) * np.cos(t), (2 + np.cos(v)) * np.sin(t), np.sin(v)])


    for n in (1, 2, 3):
        links[f"(2, {2 * n}) torus link"] = (torus_link(n, 0.0), torus_link(n, np.pi))

    for name, (a, b) in links.items():
        print(f"{name:>20}: linking number {linking_number(a, b):+.6f}")





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

 .. code-block:: none

                  unlink: linking number -0.000000
               Hopf link: linking number -1.000000
     Hopf link, reversed: linking number +1.000000
       (2, 2) torus link: linking number -1.000000
       (2, 4) torus link: linking number -2.000000
       (2, 6) torus link: linking number -3.000000




.. GENERATED FROM PYTHON SOURCE LINES 41-43

The links
---------

.. GENERATED FROM PYTHON SOURCE LINES 43-55

.. code-block:: Python


    fig = plt.figure(figsize=(12, 7))
    for i, (name, (a, b)) in enumerate(links.items()):
        ax = fig.add_subplot(2, 3, i + 1, projection="3d")
        for curve, color in ((a, "tab:blue"), (b, "tab:orange")):
            closed = np.vstack([curve, curve[:1]])
            ax.plot(*closed.T, color=color, lw=2)
        ax.set_title(f"{name}: lk = {round(linking_number(a, b))}")
        ax.set_box_aspect((1, 1, 0.6))
        ax.set_axis_off()
    fig.suptitle("Gauss (1833): the linking integral is an integer")




.. image-sg:: /api/gallery/topology/curves/images/sphx_glr_plot_01_gauss_linking_number_001.png
   :alt: Gauss (1833): the linking integral is an integer, unlink: lk = 0, Hopf link: lk = -1, Hopf link, reversed: lk = 1, (2, 2) torus link: lk = -1, (2, 4) torus link: lk = -2, (2, 6) torus link: lk = -3
   :srcset: /api/gallery/topology/curves/images/sphx_glr_plot_01_gauss_linking_number_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 0.98, 'Gauss (1833): the linking integral is an integer')



.. GENERATED FROM PYTHON SOURCE LINES 56-62

The integrand is a solid angle
------------------------------

Gauss read the integral as the solid angle swept by the direction from
one curve to the other. Refining the polygons barely changes the sum,
which stays an integer: it is exact for polygons, not an approximation.

.. GENERATED FROM PYTHON SOURCE LINES 62-69

.. code-block:: Python


    ns = [12, 24, 48, 96, 192]
    values = []
    for n in ns:
        s = np.linspace(0, 2 * np.pi, n, endpoint=False)
        values.append(linking_number(np.column_stack([np.cos(s), np.sin(s), 0 * s]), np.column_stack([1 + np.cos(s), 0 * s, np.sin(s)])))
    print("Hopf link with n-gon curves:", np.round(values, 12))




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

 .. code-block:: none

    Hopf link with n-gon curves: [-1. -1. -1. -1. -1.]





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

   **Total running time of the script:** (0 minutes 0.197 seconds)


.. _sphx_glr_download_api_gallery_topology_curves_plot_01_gauss_linking_number.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/curves/plot_01_gauss_linking_number.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_gauss_linking_number.ipynb <plot_01_gauss_linking_number.ipynb>`

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

      :download:`Download Python source code: plot_01_gauss_linking_number.py <plot_01_gauss_linking_number.py>`

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

      :download:`Download zipped: plot_01_gauss_linking_number.zip <plot_01_gauss_linking_number.zip>`


.. only:: html

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

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