.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/geometry/polygon/plot_02_heron_formula.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_02_heron_formula.py>`
        to download the full example code.

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

.. _sphx_glr_api_gallery_geometry_polygon_plot_02_heron_formula.py:


Heron's formula and needle-like triangles
===============================================

Checks Heron's formula against the shoelace formula, then shows why the
arrangement matters: for a thin "needle" triangle the textbook form
sqrt(s(s-a)(s-b)(s-c)) loses most of its digits, while Kahan's
rearrangement stays accurate.

.. GENERATED FROM PYTHON SOURCE LINES 12-17

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.geometry import heron_area, polygon_area








.. GENERATED FROM PYTHON SOURCE LINES 18-20

Agreement with the shoelace formula
-----------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-25

.. code-block:: Python


    tri = np.array([[0.0, 0.0], [4.0, 0.0], [1.0, 3.0]])
    a, b, c = (np.linalg.norm(tri[i] - tri[(i + 1) % 3]) for i in range(3))
    print(f"sides {a:.4f}, {b:.4f}, {c:.4f}: Heron {heron_area(a, b, c):.12f}, shoelace {polygon_area(tri):.12f}")





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

 .. code-block:: none

    sides 4.0000, 4.2426, 3.1623: Heron 6.000000000000, shoelace 6.000000000000




.. GENERATED FROM PYTHON SOURCE LINES 26-28

Needle triangles: textbook form vs. Kahan's arrangement
------------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 28-47

.. code-block:: Python



    def textbook(a, b, c):
        s = (a + b + c) / 2
        return np.sqrt(max(s * (s - a) * (s - b) * (s - c), 0.0))


    widths = np.logspace(-1, -12, 40)
    exact = [w / 2 * np.sqrt(1 - w * w / 4) for w in widths]  # isosceles, legs of length 1, base w
    naive_err = [abs(textbook(1.0, 1.0, w) - e) / e for w, e in zip(widths, exact)]
    kahan_err = [max(abs(heron_area(1.0, 1.0, w) - e) / e, 1e-17) for w, e in zip(widths, exact)]

    fig, ax = plt.subplots()
    ax.loglog(widths, naive_err, "o-", ms=3, label="textbook Heron")
    ax.loglog(widths, kahan_err, "s-", ms=3, label="Kahan's arrangement")
    ax.invert_xaxis()
    ax.set_xlabel("base of isosceles triangle with unit legs")
    ax.set_ylabel("relative error in area")
    ax.legend()



.. image-sg:: /api/gallery/geometry/polygon/images/sphx_glr_plot_02_heron_formula_001.png
   :alt: plot 02 heron formula
   :srcset: /api/gallery/geometry/polygon/images/sphx_glr_plot_02_heron_formula_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none


    <matplotlib.legend.Legend object at 0x1192fba10>




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

   **Total running time of the script:** (0 minutes 0.040 seconds)


.. _sphx_glr_download_api_gallery_geometry_polygon_plot_02_heron_formula.py:

.. only:: html

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

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

      :download:`Download Jupyter notebook: plot_02_heron_formula.ipynb <plot_02_heron_formula.ipynb>`

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

      :download:`Download Python source code: plot_02_heron_formula.py <plot_02_heron_formula.py>`

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

      :download:`Download zipped: plot_02_heron_formula.zip <plot_02_heron_formula.zip>`


.. only:: html

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

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