Conformal Mappings – Now with Holes!


Professor Darren Crowdy, from Imperial College, London, has solved a 140-year-old problem in mathematics by discovering Schwarz-Christoffel mapping formulas for multiply connected regions of the complex plane. Schwarz-Christoffel mappings are conformal mappings (WP, MW) from the upper-half plane to a polygon. They are used extensively in potential theory, fluid dynamics, and aircraft design, but until recently could only be applied to simply connected regions (and some doubly connected regions).

According to Crowdy (from the Times article above):

This formula is an essential piece of mathematical kit which is used the world over… Now, with my additions to it, it can be used in far more complex scenarios. In industry, for example, this mapping tool was previously inadequate if a piece of metal or other material was not uniform – for instance, if it contained parts of a different material, or had holes. With my extensions to this formula, you can take account of these differences and map them on to a simple disk shape for analysis in the same way as you can with less complex shapes without any of the holes.

There is a longer discussion of the mathematics behind the Schwarz-Christoffel formula, including some of the history of its solution, at SIAM. For some great visualizations of conformal mappings, see the Virtual Math Museum, as well as the link above.

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