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

.. only:: html

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

        Click :ref:`here <sphx_glr_download_tutorials_geodetic_to_geocentric.py>`
        to download the full example code

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

.. _sphx_glr_tutorials_geodetic_to_geocentric.py:


.. _geodetic_to_geocentric:

Coordinate conversions
======================

Both geocentric coordinate system and geodetic coordinate system are geographic
coordinate systems centred at the mass of the earth. In geocentric coordinate
system, the earth is modelled as a sphere. The geodetic coordinate system is
based on the reference ellipsoid considering the centrifugal force due to
the earth's rotation.

The major difference between these two systems is the way how they define
latitude and height. In geocentric coordinate system, the geocentric latitude
is defined by the angle between the plane the equator and the line from earth
centre to a given point. The distance from the point to the earth centre is
geocentric radius :`R`. In geodetic coordinate system, the geodetic latitude is
the angle between the plane of the equator and the surface normal at a point
on the ellipsoid. The ellipsoid height :math:`h` is the height above the
ellipsoid surface, it also normal to that surface.

.. figure:: ../_static/coordinate.png
    :alt: Depiction of geocentric and geodetic systems with associated coordinates.

    Geocentric and Geodetic Coordinates Systems

Note, the flattening of the shown ellipsoid is larger than the real earth.
The real difference between geodetic latitude and geocentric latitude at
the earth ellipsoid surface is less than 0.2 degrees.

The calculation is performed by the :meth:`boule.Ellipsoid.geodetic_to_spherical`
method.

This example will show you how to convert geodetic latitude and height into
geocentric latitude and radius. We use :ref:`WGS84 <wgs84>`  ellipsoid in here.

.. GENERATED FROM PYTHON SOURCE LINES 44-77

.. code-block:: default


    import numpy as np
    import matplotlib.pyplot as plt
    import boule as bl


    latitude = np.linspace(-90, 90, 100)
    longitude = 0
    # ellipsoid surface
    Height1 = 0
    # ICESat-2 orbit height
    Height2 = 481000
    SP_Cor1 = bl.WGS84.geodetic_to_spherical(longitude, latitude, Height1)
    SP_Cor2 = bl.WGS84.geodetic_to_spherical(longitude, latitude, Height2)

    fig, (axs1, axs2, axs3) = plt.subplots(3, 1, figsize=(10, 15))

    axs1.plot(latitude, SP_Cor1[1], "-k")
    axs1.set_title("Geodetic latitude to geocentric latitude at ellipsoid surface")
    axs1.set(xlabel="Geodetic Latitude", ylabel="Geocentric Latitude")

    axs2.plot(latitude, SP_Cor1[2], "-k")
    axs2.set_title("Geocentric radius at ellipsoid surface")
    axs2.set(xlabel="Geodetic Latitude", ylabel="Geocentric Radius (m)")

    axs3.plot(latitude, latitude - SP_Cor1[1], "-k")
    axs3.set_title(
        "Difference between geodetic latitude and geocentric latitude at ellipsoid surface"
    )
    axs3.set(xlabel="Geodetic Latitude", ylabel="Difference (Degree)")

    plt.show()




.. image-sg:: /tutorials/images/sphx_glr_geodetic_to_geocentric_001.png
   :alt: Geodetic latitude to geocentric latitude at ellipsoid surface, Geocentric radius at ellipsoid surface, Difference between geodetic latitude and geocentric latitude at ellipsoid surface
   :srcset: /tutorials/images/sphx_glr_geodetic_to_geocentric_001.png
   :class: sphx-glr-single-img





.. GENERATED FROM PYTHON SOURCE LINES 78-83

The geocentric latitude is a function of geodetic latitude and ellipsoid height.
The influence of ellipsoid height change on the geocentric latitude varies with
the geodetic latitude. Consider different ellipsoid heights (ICESat-2 orbit
height vs ellipsoid surface), this difference could be 0.013 degrees. Similarly,
the Geocentric radius change could be 2.5 meters.

.. GENERATED FROM PYTHON SOURCE LINES 83-95

.. code-block:: default


    fig, (axs1, axs2) = plt.subplots(2, 1, figsize=(10, 10))

    axs1.plot(latitude, SP_Cor2[1] - SP_Cor1[1], "-k")
    axs1.set_title("Geocentric latitude changes at different ellipsoid heights")
    axs1.set(xlabel="Geodetic Latitude", ylabel="Difference (Degree)")

    axs2.plot(latitude, SP_Cor2[2] - SP_Cor1[2], "-k")
    axs2.set_title("Geocentric radius changes at different ellipsoid height")
    axs2.set(xlabel="Geodetic Latitude", ylabel="Difference (m)")

    plt.show()



.. image-sg:: /tutorials/images/sphx_glr_geodetic_to_geocentric_002.png
   :alt: Geocentric latitude changes at different ellipsoid heights, Geocentric radius changes at different ellipsoid height
   :srcset: /tutorials/images/sphx_glr_geodetic_to_geocentric_002.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** ( 0 minutes  0.763 seconds)


.. _sphx_glr_download_tutorials_geodetic_to_geocentric.py:


.. only :: html

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



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

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



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

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


.. only:: html

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

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