Schwarz-Christoffel transformation

The Schwarz-Christoffel transformation (abbreviation SCT) is a mathematical mapping rule which allows a standardized mathematical area to be mapped onto any polygon with precise angles. In special cases, circles (continuous, known kink angle) can also be mapped in this way.

This means that mechanical and electrical calculations can be carried out more easily on bodies and surfaces with complex boundaries, because the relationships in the original bodies are known and can be easily represented. The transformation goes back to the two German mathematicians Hermann Amandus Schwarz and Elwin Bruno Christoffel .

formula

A mapping of the real number axis on any line, with inside / kink angles and original images of the corner points is given by: ${\ displaystyle \ pi \ alpha _ {1}, \ pi \ alpha _ {2}, ..., \ pi \ alpha _ {n}}$ ${\ displaystyle a_ {1}, a_ {2}, ..., a_ {n}}$

{\ displaystyle f (z) = C \ int _ {0} ^ {z} (\ zeta -a_ {1}) ^ {\ alpha _ {1} -1} (\ zeta -a_ {2}) ^ { \ alpha _ {2} -1} ... (\ zeta -a_ {n}) ^ {\ alpha _ {n} -1} d \ zeta + C ^ {'}}

or alternatively through the DGL:

{\ displaystyle {\ frac {f '' (z)} {f '(z)}} = \ left \ {{\ begin {array} {ll} \ sum _ {k = 1} ^ {n} {\ frac {\ alpha _ {k} -1} {z-a_ {k}}} & {\ text {if all}} a_ {k} \ neq \ infty \\\ sum _ {k = 1} ^ {n -1} {\ frac {\ alpha _ {k} -1} {z-a_ {k}}} & {\ text {falls}} a_ {n} = \ infty \ end {array}} \ right .. }

When the line is closed, i.e. in the case of an n-gon , the upper half -plane is mapped onto the interior of the n-gon. In the case of a triangle, because of the conformal self-mapping of the upper half-plane (the Möbius transformations), which are completely defined by three entries, the archetypes can be chosen at will. ${\ displaystyle \ Pi}$ ${\ displaystyle \ mathbb {H}}$ ${\ displaystyle n = 3}$

Graphic interpretation

Example of a level transformation using the SCT

The two selected kink points on the X axis are mapped to the corner points of the starting area. In this way, the upper half-plane with a known potential profile is transferred into the bent structure to be analyzed. The choice of points is not fundamentally important, it only influences the representation of the transformation. By forming the inverse transformation and solving it, the complicated curves of the electrical potential can be represented directly.

Web links

Schwarz-Christoffel transformation on PlanetMath
John H. Mathews and Russell W. Howell: Module for The Schwarz-Christoffel Transformation