.. _sphx_glr_api_gallery_complex_analysis_conformal_maps:

Conformal maps
--------------

Möbius transformations, the Riemann mapping theorem, and the Joukowski airfoil.


.. raw:: html

  <div id='sg-tag-list' class='sphx-glr-tag-list'></div>


.. raw:: html

    <div class="sphx-glr-thumbnails">

.. thumbnail-parent-div-open

.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="Riemann (1851) asserted that every simply connected proper subdomain of the plane can be mapped conformally onto the unit disk. For the upper half-plane the map is explicit: the Cayley transform w = (z - i)/(z + i), a Möbius transformation. The image of the Cartesian grid is a family of circles that still meet at right angles, because conformal maps preserve angles.">

.. only:: html

  .. image:: /api/gallery/complex_analysis/conformal_maps/images/thumb/sphx_glr_plot_01_riemann_mapping_theorem_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/conformal_maps/plot_01_riemann_mapping_theorem`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Riemann's mapping theorem: the upper half-plane mapped conformally onto the disk</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="J(z) = z + 1/z flattens the unit circle onto [-2, 2]. A circle through z = 1 that encloses -1 maps to an airfoil with a sharp trailing edge, where J&#x27;(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.">

.. only:: html

  .. image:: /api/gallery/complex_analysis/conformal_maps/images/thumb/sphx_glr_plot_02_joukowski_airfoil_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/conformal_maps/plot_02_joukowski_airfoil`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">The Joukowski map: turning circles into airfoils</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="August Ferdinand Möbius&#x27;s 1855 theory of Kreisverwandtschaft (&quot;circle relationship&quot;) studied the maps w = (az + b)/(cz + d), ad - bc \ne 0. Every one of them sends circles and lines to circles and lines, where a line is a circle through \infty. This script maps a family of circles and checks that each image is again a circle by fitting one to it.">

.. only:: html

  .. image:: /api/gallery/complex_analysis/conformal_maps/images/thumb/sphx_glr_plot_03_mobius_circles_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/conformal_maps/plot_03_mobius_circles`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Möbius transformations map circles to circles</div>
    </div>


.. raw:: html

    <div class="sphx-glr-thumbcontainer" tooltip="If f maps the unit disk into itself with f(0) = 0, then |f(z)| \le |z| and |f&#x27;(0)| \le 1, with equality only for rotations f(z) = e^{i\theta}z (Schwarz 1869; Carathéodory gave the general statement in 1912). Blaschke products, the model self-maps of the disk, obey it with room to spare.">

.. only:: html

  .. image:: /api/gallery/complex_analysis/conformal_maps/images/thumb/sphx_glr_plot_04_schwarz_lemma_thumb.png
    :alt:

  :doc:`/api/gallery/complex_analysis/conformal_maps/plot_04_schwarz_lemma`

.. raw:: html

      <div class="sphx-glr-thumbnail-title">Schwarz's lemma: self-maps of the disk fixing 0 cannot expand</div>
    </div>


.. thumbnail-parent-div-close

.. raw:: html

    </div>


.. toctree::
   :hidden:

   /api/gallery/complex_analysis/conformal_maps/plot_01_riemann_mapping_theorem
   /api/gallery/complex_analysis/conformal_maps/plot_02_joukowski_airfoil
   /api/gallery/complex_analysis/conformal_maps/plot_03_mobius_circles
   /api/gallery/complex_analysis/conformal_maps/plot_04_schwarz_lemma