.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "auto_examples/a_coordinate_systems/plot_homoeoidal_coordinates.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

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

        :ref:`Go to the end <sphx_glr_download_auto_examples_a_coordinate_systems_plot_homoeoidal_coordinates.py>`
        to download the full example code.

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

.. _sphx_glr_auto_examples_a_coordinate_systems_plot_homoeoidal_coordinates.py:


======================================================
Oblate Homoeoidal Coordinates: Effect of Eccentricity
======================================================

Visualize how the radial coordinate lines (R-contours) in the OblateHomoeoidalCoordinateSystem
change as a function of eccentricity.

.. GENERATED FROM PYTHON SOURCE LINES 9-11

.. code-block:: Python

    import matplotlib.pyplot as plt








.. GENERATED FROM PYTHON SOURCE LINES 12-21

In this example, we'll showcase a more complex coordinate system:
:class:`~coordinates.coordinate_systems.OblateHomoeoidalCoordinateSystem`.

The oblate homoeoidal coordinate system is a fully curvilinear coordinate system
(non-diagonal metric) which inherits the angular coordinates of spherical coordinates
but instead relies on a modified effective radius composed of con-centric ellipsoids.

In this example, we'll show off how to make these coordinate systems
and then display some of its properties.

.. GENERATED FROM PYTHON SOURCE LINES 21-25

.. code-block:: Python

    import numpy as np

    from pymetric import OblateHomoeoidalCoordinateSystem








.. GENERATED FROM PYTHON SOURCE LINES 26-29

To visualize, we'll use an *x-z* grid and then convert it into the
relevant coordinate systems with different eccentricities. We can then
plot contours for the effective radius :math:`\xi`.

.. GENERATED FROM PYTHON SOURCE LINES 29-35

.. code-block:: Python


    # Create the cartesian grid.
    x = np.linspace(-1.5, 1.5, 200)
    z = np.linspace(-1.5, 1.5, 200)
    X, Z = np.meshgrid(x, z)








.. GENERATED FROM PYTHON SOURCE LINES 36-40

The class requires an eccentricity parameter when
initialized, so we'll iterate through each of them,
create a coordinate system class, and then perform
the conversions.

.. GENERATED FROM PYTHON SOURCE LINES 40-64

.. code-block:: Python

    eccentricities = [0.0, 0.1, 0.3, 0.6, 0.9, 0.99]

    fig, axes = plt.subplots(
        int(np.ceil(len(eccentricities) / 3)), 3, sharex=True, sharey=True
    )

    for i, ecc in enumerate(eccentricities):
        ax = axes.ravel()[i]
        csys = OblateHomoeoidalCoordinateSystem(ecc=ecc)

        # Convert Cartesian (x, z) to native coordinates (λ, μ, φ)
        Lambda, Mu, Phi = csys.from_cartesian(X, 0, Z)

        # Plot λ contours (these correspond to elliptical shells)
        contour = ax.contour(X, Z, Lambda, levels=15, cmap="viridis")
        ax.set_title(f"$\\varepsilon = {ecc}$")
        ax.set_aspect("equal")
        ax.set_xlabel("x")
        if i == 0:
            ax.set_ylabel("z")

    fig.suptitle(r"Oblate Homoeoidal Coordinate Contours ($\lambda$-lines)", fontsize=14)
    plt.tight_layout()
    plt.show()



.. image-sg:: /auto_examples/a_coordinate_systems/images/sphx_glr_plot_homoeoidal_coordinates_001.png
   :alt: Oblate Homoeoidal Coordinate Contours ($\lambda$-lines), $\varepsilon = 0.0$, $\varepsilon = 0.1$, $\varepsilon = 0.3$, $\varepsilon = 0.6$, $\varepsilon = 0.9$, $\varepsilon = 0.99$
   :srcset: /auto_examples/a_coordinate_systems/images/sphx_glr_plot_homoeoidal_coordinates_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 11.626 seconds)


.. _sphx_glr_download_auto_examples_a_coordinate_systems_plot_homoeoidal_coordinates.py:

.. only:: html

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

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

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

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

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

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

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


.. only:: html

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

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