.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/geometry/polygon/plot_03_pick_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_geometry_polygon_plot_03_pick_theorem.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_geometry_polygon_plot_03_pick_theorem.py:


Pick's theorem: area from lattice points
==============================================

Counts the lattice points inside and on the boundary of polygons with
integer vertices, and checks Pick's formula A = I + B/2 - 1 against the
shoelace area.

.. GENERATED FROM PYTHON SOURCE LINES 11-16

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.geometry import lattice_point_counts, point_in_polygon








.. GENERATED FROM PYTHON SOURCE LINES 17-19

Several lattice polygons
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 19-29

.. code-block:: Python


    polygons = {
        "rectangle": [[0, 0], [4, 0], [4, 3], [0, 3]],
        "triangle": [[0, 0], [7, 2], [3, 6]],
        "non-convex": [[0, 0], [6, 0], [6, 5], [3, 2], [0, 5]],
    }
    for name, vertices in polygons.items():
        r = lattice_point_counts(vertices)
        print(f"{name:10s}: I = {r.interior:2d}, B = {r.boundary:2d}, I + B/2 - 1 = {r.pick_area:5.1f}, shoelace area = {r.area:5.1f}")





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

 .. code-block:: none

    rectangle : I =  6, B = 14, I + B/2 - 1 =  12.0, shoelace area =  12.0
    triangle  : I = 15, B =  8, I + B/2 - 1 =  18.0, shoelace area =  18.0
    non-convex: I = 11, B = 22, I + B/2 - 1 =  21.0, shoelace area =  21.0




.. GENERATED FROM PYTHON SOURCE LINES 30-32

Picture of the non-convex case
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 32-42

.. code-block:: Python


    v = np.array(polygons["non-convex"], dtype=float)
    fig, ax = plt.subplots()
    ax.fill(*v.T, alpha=0.3)
    for x in range(0, 7):
        for y in range(0, 6):
            inside = point_in_polygon(np.array([x, y], dtype=float), v)
            ax.plot(x, y, "o", color="tab:green" if inside else "0.8", ms=5)
    ax.set_aspect("equal")
    ax.set_title("Pick: area = interior points + boundary points / 2 - 1")



.. image-sg:: /api/gallery/geometry/polygon/images/sphx_glr_plot_03_pick_theorem_001.png
   :alt: Pick: area = interior points + boundary points / 2 - 1
   :srcset: /api/gallery/geometry/polygon/images/sphx_glr_plot_03_pick_theorem_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    Text(0.5, 1.0, 'Pick: area = interior points + boundary points / 2 - 1')




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

   **Total running time of the script:** (0 minutes 0.025 seconds)


.. _sphx_glr_download_api_gallery_geometry_polygon_plot_03_pick_theorem.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_03_pick_theorem.ipynb <plot_03_pick_theorem.ipynb>`

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

      :download:`Download Python source code: plot_03_pick_theorem.py <plot_03_pick_theorem.py>`

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

      :download:`Download zipped: plot_03_pick_theorem.zip <plot_03_pick_theorem.zip>`


.. only:: html

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

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