.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/complex_analysis/conformal_maps/plot_03_mobius_circles.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_complex_analysis_conformal_maps_plot_03_mobius_circles.py>`
        to download the full example code or to run this example in your browser via JupyterLite.

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

.. _sphx_glr_api_gallery_complex_analysis_conformal_maps_plot_03_mobius_circles.py:


Möbius transformations map circles to circles
=============================================

August Ferdinand Möbius's 1855 theory of *Kreisverwandtschaft*
("circle relationship") studied the maps
:math:`w = (az + b)/(cz + d)`, :math:`ad - bc \ne 0`. Every one of them
sends circles and lines to circles and lines, where a line is a circle
through :math:`\infty`. This script maps a family of circles and checks
that each image is again a circle by fitting one to it.

.. GENERATED FROM PYTHON SOURCE LINES 14-32

.. code-block:: Python

    import matplotlib.pyplot as plt
    import numpy as np

    from mathematicskit.complex_analysis import circle_contour, mobius_transform

    a, b, c, d = 1.0, -0.5j, 0.6, 1.0 + 0.3j


    def fit_circle(points):
        """Least-squares circle x^2 + y^2 + Dx + Ey + F = 0; returns center, radius, max residual."""
        x, y = points.real, points.imag
        A = np.column_stack([x, y, np.ones_like(x)])
        D, E, F = np.linalg.lstsq(A, -(x**2 + y**2), rcond=None)[0]
        center = complex(-D / 2, -E / 2)
        radius = np.sqrt(abs(center) ** 2 - F)
        return center, radius, np.max(np.abs(np.abs(points - center) - radius))









.. GENERATED FROM PYTHON SOURCE LINES 33-35

Circles in the z-plane and their images
---------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 35-49

.. code-block:: Python


    fig, (ax0, ax1) = plt.subplots(1, 2, figsize=(11, 5))
    for k, (center, radius) in enumerate([(0.0, 0.5), (0.5 + 0.5j, 0.3), (-0.6, 0.4), (0.2 - 0.6j, 0.25)]):
        z = circle_contour(center, radius).points(400)
        w = mobius_transform(z, a, b, c, d)
        color = f"C{k}"
        ax0.plot(z.real, z.imag, color)
        ax1.plot(w.real, w.imag, color)
        w_center, w_radius, residual = fit_circle(w)
        print(f"circle {k}: image center {w_center:.4f}, radius {w_radius:.4f}, max deviation {residual:.1e}")
    for ax, title in ((ax0, "z-plane"), (ax1, "w = (az + b)/(cz + d)")):
        ax.set_aspect("equal")
        ax.set_title(title)




.. image-sg:: /api/gallery/complex_analysis/conformal_maps/images/sphx_glr_plot_03_mobius_circles_001.png
   :alt: z-plane, w = (az + b)/(cz + d)
   :srcset: /api/gallery/complex_analysis/conformal_maps/images/sphx_glr_plot_03_mobius_circles_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    circle 0: image center -0.3000-0.5000j, radius 0.5831, max deviation 8.9e-16
    circle 1: image center 0.2954-0.1487j, radius 0.1734, max deviation 4.7e-16
    circle 2: image center -1.4253-0.3167j, radius 1.0554, max deviation 1.1e-15
    circle 3: image center 0.2044-0.9875j, radius 0.2360, max deviation 2.1e-15




.. GENERATED FROM PYTHON SOURCE LINES 50-52

A circle through the pole -d/c becomes a line
---------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 52-62

.. code-block:: Python


    pole = -d / c
    theta = np.linspace(0.0, 2.0 * np.pi, 401)[1:-1]  # theta = 0 is the pole itself
    z = pole + 0.5 * (1.0 - np.exp(1j * theta))
    w = mobius_transform(z, a, b, c, d)
    p, q = w[100], w[300]  # two finite image points fix the line
    distance = np.abs(((w - p) * np.conj(q - p)).imag) / abs(q - p)
    print(f"max distance of the images from the line through two of them: {np.max(distance / np.maximum(1.0, np.abs(w))):.1e} (relative)")

    plt.show()




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

 .. code-block:: none

    max distance of the images from the line through two of them: 1.8e-14 (relative)





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

   **Total running time of the script:** (0 minutes 0.117 seconds)


.. _sphx_glr_download_api_gallery_complex_analysis_conformal_maps_plot_03_mobius_circles.py:

.. only:: html

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

    .. container:: lite-badge

      .. image:: images/jupyterlite_badge_logo.svg
        :target: ../../../../lite/lab/index.html?path=api/gallery/complex_analysis/conformal_maps/plot_03_mobius_circles.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_03_mobius_circles.ipynb <plot_03_mobius_circles.ipynb>`

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

      :download:`Download Python source code: plot_03_mobius_circles.py <plot_03_mobius_circles.py>`

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

      :download:`Download zipped: plot_03_mobius_circles.zip <plot_03_mobius_circles.zip>`


.. only:: html

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

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