.. 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_02_joukowski_airfoil.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_02_joukowski_airfoil.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_02_joukowski_airfoil.py:


The Joukowski map: turning circles into airfoils
================================================

:math:`J(z) = z + 1/z` flattens the unit circle onto :math:`[-2, 2]`.
A circle through :math:`z = 1` that encloses :math:`-1` maps to an
airfoil with a sharp trailing edge, where :math:`J'(1) = 0` and the map
stops being conformal. Joukowski used this in 1910 to carry the
solvable flow around a cylinder over to a wing profile.

.. GENERATED FROM PYTHON SOURCE LINES 13-17

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.complex_analysis import circle_contour, complex_derivative, joukowski_map








.. GENERATED FROM PYTHON SOURCE LINES 18-20

Circles and their images
------------------------

.. GENERATED FROM PYTHON SOURCE LINES 20-35

.. code-block:: Python


    centers = {"unit circle": 0.0, "shifted left": -0.1, "shifted left and up (airfoil)": -0.1 + 0.15j}
    fig, axes = plt.subplots(1, 2, figsize=(11, 4.5))
    for name, c in centers.items():
        circle = circle_contour(c, abs(1 - c))  # every circle passes through z = 1
        z = circle.points(600)
        w = joukowski_map(z)
        axes[0].plot(z.real, z.imag, label=name)
        axes[1].plot(w.real, w.imag, label=name)
    axes[0].plot([1, -1], [0, 0], "kx")
    for ax, title in zip(axes, ("z-plane", "w = z + 1/z"), strict=False):
        ax.set_aspect("equal")
        ax.set_title(title)
        ax.legend(fontsize=8)
    print("J'(1) =", complex_derivative(joukowski_map, 1.0), "(the trailing edge is a critical point)")



.. image-sg:: /api/gallery/complex_analysis/conformal_maps/images/sphx_glr_plot_02_joukowski_airfoil_001.png
   :alt: z-plane, w = z + 1/z
   :srcset: /api/gallery/complex_analysis/conformal_maps/images/sphx_glr_plot_02_joukowski_airfoil_001.png
   :class: sphx-glr-single-img


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

 .. code-block:: none

    J'(1) = 0j (the trailing edge is a critical point)





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

   **Total running time of the script:** (0 minutes 0.127 seconds)


.. _sphx_glr_download_api_gallery_complex_analysis_conformal_maps_plot_02_joukowski_airfoil.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_02_joukowski_airfoil.ipynb
        :alt: Launch JupyterLite
        :width: 150 px

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

      :download:`Download Jupyter notebook: plot_02_joukowski_airfoil.ipynb <plot_02_joukowski_airfoil.ipynb>`

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

      :download:`Download Python source code: plot_02_joukowski_airfoil.py <plot_02_joukowski_airfoil.py>`

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

      :download:`Download zipped: plot_02_joukowski_airfoil.zip <plot_02_joukowski_airfoil.zip>`


.. only:: html

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

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