.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/geometry/surfaces/plot_01_theorema_egregium.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_geometry_surfaces_plot_01_theorema_egregium.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_geometry_surfaces_plot_01_theorema_egregium.py:


Gauss's Theorema Egregium: curvature you can measure from inside
======================================================================

Computes the Gaussian curvature K of a sphere, a cylinder, and a torus.
The cylinder bends in space but has K = 0, because it can be unrolled
flat without stretching, while no piece of a sphere can. Integrating K
over the whole surface gives 4 pi for the sphere and 0 for the torus,
as the Gauss-Bonnet theorem predicts.

.. GENERATED FROM PYTHON SOURCE LINES 13-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np
    from scipy import integrate

    from mathematicskit.geometry import cylinder_surface, sphere_surface, surface_curvature, torus_surface








.. GENERATED FROM PYTHON SOURCE LINES 20-22

Curvature of three surfaces
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 22-38

.. code-block:: Python


    cases = {
        "sphere (R = 1)": surface_curvature(sphere_surface(1.0), np.linspace(1e-3, np.pi - 1e-3, 201), np.linspace(0, 2 * np.pi, 201)),
        "cylinder (R = 1)": surface_curvature(cylinder_surface(1.0), np.linspace(-1, 1, 101), np.linspace(0, 2 * np.pi, 201)),
        "torus (R = 2, r = 0.7)": surface_curvature(torus_surface(2.0, 0.7), np.linspace(0, 2 * np.pi, 201), np.linspace(0, 2 * np.pi, 201)),
    }
    fig = plt.figure(figsize=(13, 4))
    for k, (name, result) in enumerate(cases.items(), start=1):
        ax = fig.add_subplot(1, 3, k, projection="3d")
        K = result.gaussian
        colors = plt.cm.coolwarm((K - K.min()) / (np.ptp(K) + 1e-12)) if np.ptp(K) > 1e-9 else plt.cm.coolwarm(np.full(K.shape, 0.5))
        ax.plot_surface(*result.points, facecolors=colors, rstride=4, cstride=4, linewidth=0)
        ax.set_title(f"{name}\nK from {K.min():+.3f} to {K.max():+.3f}", fontsize=9)
        ax.set_box_aspect((1, 1, 1))
        ax.axis("off")




.. image-sg:: /api/gallery/geometry/surfaces/images/sphx_glr_plot_01_theorema_egregium_001.png
   :alt: sphere (R = 1) K from +0.999 to +1.001, cylinder (R = 1) K from -0.000 to +0.000, torus (R = 2, r = 0.7) K from -1.099 to +0.529
   :srcset: /api/gallery/geometry/surfaces/images/sphx_glr_plot_01_theorema_egregium_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 39-41

Total curvature (Gauss-Bonnet)
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 41-48

.. code-block:: Python


    for name, result in cases.items():
        if name.startswith("cylinder"):
            continue
        total = integrate.trapezoid(integrate.trapezoid(result.gaussian * result.area_element, result.v, axis=1), result.u)
        print(f"{name:22s}: integral of K dA = {total:+.5f}")
    print(f"{'4 pi':22s}  = {4 * np.pi:+.5f}")




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

 .. code-block:: none

    sphere (R = 1)        : integral of K dA = +12.56352
    torus (R = 2, r = 0.7): integral of K dA = -0.00002
    4 pi                    = +12.56637





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

   **Total running time of the script:** (0 minutes 0.144 seconds)


.. _sphx_glr_download_api_gallery_geometry_surfaces_plot_01_theorema_egregium.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_01_theorema_egregium.ipynb <plot_01_theorema_egregium.ipynb>`

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

      :download:`Download Python source code: plot_01_theorema_egregium.py <plot_01_theorema_egregium.py>`

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

      :download:`Download zipped: plot_01_theorema_egregium.zip <plot_01_theorema_egregium.zip>`


.. only:: html

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

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