.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/geometry/polygon/plot_01_euclid_equal_areas.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_polygon_plot_01_euclid_equal_areas.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_geometry_polygon_plot_01_euclid_equal_areas.py:


Euclid's Elements, Book I: equal areas between parallels
==============================================================

Euclid's theory of area (*Elements* I.35-41) rests on two facts:
parallelograms on the same base and between the same parallels are
equal in area (I.35), and so are triangles (I.37), each triangle being
half of the parallelogram (I.41). Sliding the top side along its
parallel line changes every shape but, as the shoelace formula in
:func:`~mathematicskit.geometry.polygon_area` confirms, never the area.

.. GENERATED FROM PYTHON SOURCE LINES 14-19

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.geometry import polygon_area, polygon_centroid








.. GENERATED FROM PYTHON SOURCE LINES 20-24

Parallelograms on one base between two parallels (I.35)
-------------------------------------------------------
The base runs from (0, 0) to (4, 0); the opposite side lies on the
parallel line y = 2, shifted sideways by different amounts.

.. GENERATED FROM PYTHON SOURCE LINES 24-33

.. code-block:: Python


    base, height = 4.0, 2.0
    shifts = [0.0, 2.5, 6.0]
    parallelograms = [np.array([[0, 0], [base, 0], [base + s, height], [s, height]]) for s in shifts]
    for s, quad in zip(shifts, parallelograms):
        cx, cy = polygon_centroid(quad)
        print(f"shift {s:3.1f}: area {polygon_area(quad):.6f}, centroid ({cx:.2f}, {cy:.2f})")
    print(f"base x height = {base * height:.6f}")





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

 .. code-block:: none

    shift 0.0: area 8.000000, centroid (2.00, 1.00)
    shift 2.5: area 8.000000, centroid (3.25, 1.00)
    shift 6.0: area 8.000000, centroid (5.00, 1.00)
    base x height = 8.000000




.. GENERATED FROM PYTHON SOURCE LINES 34-36

Triangles on the same base, half the parallelogram (I.37, I.41)
---------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 36-42

.. code-block:: Python


    apexes = [0.0, 3.0, 7.0]
    triangles = [np.array([[0, 0], [base, 0], [a, height]]) for a in apexes]
    for a, tri in zip(apexes, triangles):
        print(f"apex at x = {a:3.1f}: area {polygon_area(tri):.6f} (half of {base * height:.1f})")





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

 .. code-block:: none

    apex at x = 0.0: area 4.000000 (half of 8.0)
    apex at x = 3.0: area 4.000000 (half of 8.0)
    apex at x = 7.0: area 4.000000 (half of 8.0)




.. GENERATED FROM PYTHON SOURCE LINES 43-47

The figures
-----------------------------------------------------
Every centroid lies on the midline y = h/2 for the parallelograms and
y = h/3 for the triangles: the shapes shear, their areas do not change.

.. GENERATED FROM PYTHON SOURCE LINES 47-60

.. code-block:: Python


    fig, (ax1, ax2) = plt.subplots(2, 1, figsize=(9, 4.5), sharex=True)
    colors = ["tab:blue", "tab:orange", "tab:green"]
    for ax, shapes, title in [(ax1, parallelograms, "I.35: parallelograms"), (ax2, triangles, "I.37: triangles")]:
        ax.axhline(0, color="k", lw=0.8)
        ax.axhline(height, color="k", lw=0.8, ls="--")
        for shape, color in zip(shapes, colors):
            ax.fill(*shape.T, color=color, alpha=0.3, ec=color, lw=1.5)
            ax.plot(*polygon_centroid(shape), "o", color=color)
            ax.text(*polygon_centroid(shape), f"  {polygon_area(shape):.1f}", color=color, va="center")
        ax.set_aspect("equal")
        ax.set_title(f"{title} on the same base between the same parallels")
    plt.show()



.. image-sg:: /api/gallery/geometry/polygon/images/sphx_glr_plot_01_euclid_equal_areas_001.png
   :alt: I.35: parallelograms on the same base between the same parallels, I.37: triangles on the same base between the same parallels
   :srcset: /api/gallery/geometry/polygon/images/sphx_glr_plot_01_euclid_equal_areas_001.png
   :class: sphx-glr-single-img






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

   **Total running time of the script:** (0 minutes 0.043 seconds)


.. _sphx_glr_download_api_gallery_geometry_polygon_plot_01_euclid_equal_areas.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_euclid_equal_areas.ipynb <plot_01_euclid_equal_areas.ipynb>`

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

      :download:`Download Python source code: plot_01_euclid_equal_areas.py <plot_01_euclid_equal_areas.py>`

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

      :download:`Download zipped: plot_01_euclid_equal_areas.zip <plot_01_euclid_equal_areas.zip>`


.. only:: html

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

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