.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/plate_tectonics/plot_01_euler_rotation_theorem.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_plate_tectonics_plot_01_euler_rotation_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_plate_tectonics_plot_01_euler_rotation_theorem.py:


Euler's rotation theorem (1776)
===============================

Any motion of a rigid cap on a sphere is a rotation about an axis through the
centre. The velocity of every point is the cross product of the Euler vector
and its position, so it is tangent to the sphere and vanishes at the pole. The
pole and rate here are synthetic.

.. GENERATED FROM PYTHON SOURCE LINES 10-46



.. image-sg:: /api/gallery/plate_tectonics/images/sphx_glr_plot_01_euler_rotation_theorem_001.png
   :alt: Rotation of 1 degree per million years
   :srcset: /api/gallery/plate_tectonics/images/sphx_glr_plot_01_euler_rotation_theorem_001.png
   :class: sphx-glr-single-img





.. code-block:: Python


    import matplotlib.pyplot as plt
    import numpy as np

    from earthsciencekit.constants import EARTH_RADIUS
    from earthsciencekit.plate_tectonics import angular_speed, euler_velocity_x, euler_velocity_y, euler_velocity_z
    from earthsciencekit.units import convert

    rate = angular_speed(convert(1.0, "degree", "rad"), convert(1e6, "year", "s"))
    pole_lat, pole_lon = np.deg2rad(50.0), np.deg2rad(-30.0)
    omega = rate * np.array([np.cos(pole_lat) * np.cos(pole_lon), np.cos(pole_lat) * np.sin(pole_lon), np.sin(pole_lat)])

    lon, lat = np.meshgrid(np.deg2rad(np.arange(-180, 181, 20)), np.deg2rad(np.arange(-75, 76, 15)))
    r = EARTH_RADIUS * np.stack([np.cos(lat) * np.cos(lon), np.cos(lat) * np.sin(lon), np.sin(lat)])
    v = np.stack(
        [
            euler_velocity_x(omega[1], omega[2], r[1], r[2]),
            euler_velocity_y(omega[2], omega[0], r[2], r[0]),
            euler_velocity_z(omega[0], omega[1], r[0], r[1]),
        ]
    )
    np.testing.assert_allclose(np.sum(v * r, axis=0), 0, atol=1e-9)
    pole = EARTH_RADIUS * omega / rate
    assert np.allclose(
        [euler_velocity_x(omega[1], omega[2], pole[1], pole[2]), euler_velocity_z(omega[0], omega[1], pole[0], pole[1])], 0, atol=1e-20
    )

    east = -np.sin(lon) * v[0] + np.cos(lon) * v[1]
    north = -np.sin(lat) * np.cos(lon) * v[0] - np.sin(lat) * np.sin(lon) * v[1] + np.cos(lat) * v[2]
    fig, ax = plt.subplots(figsize=(8, 4.2))
    ax.quiver(np.rad2deg(lon), np.rad2deg(lat), convert(east, "m/s", "cm/year"), convert(north, "m/s", "cm/year"), color="tab:blue")
    ax.plot(np.rad2deg(pole_lon), np.rad2deg(pole_lat), "r*", ms=14, label="Euler pole")
    ax.set(xlabel="longitude [degree]", ylabel="latitude [degree]", title="Rotation of 1 degree per million years")
    ax.legend(loc="lower left")

    plt.show()


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

   **Total running time of the script:** (0 minutes 0.101 seconds)


.. _sphx_glr_download_api_gallery_plate_tectonics_plot_01_euler_rotation_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/plate_tectonics/plot_01_euler_rotation_theorem.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_01_euler_rotation_theorem.ipynb <plot_01_euler_rotation_theorem.ipynb>`

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

      :download:`Download Python source code: plot_01_euler_rotation_theorem.py <plot_01_euler_rotation_theorem.py>`

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

      :download:`Download zipped: plot_01_euler_rotation_theorem.zip <plot_01_euler_rotation_theorem.zip>`


.. only:: html

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

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