Schwarz–Christoffel mapping

The Black - Christoffel transformation (abbreviated SCT) is a mathematical mapping rule, which allows a standardized mathematical field angle exactly mapped onto an arbitrary polygon. In the special case can thus also circles (continuous, well-known kink angle) depict.

This allows mechanical and electrical complicated calculations limited bodies and surfaces are more easily performed because the conditions known in the original specimens and can be easily displayed. The transformation goes back to the two German mathematician Hermann Amandus Schwarz and Elwin Bruno Christoffel.

Formula

The function

Forms the real axis from a polyline, which makes a bend to the angle at each of the points. Applies the polyline, so limited a polygon and forms the upper half-plane conformally onto the interior from.

Graphical interpretation

The two selected breakpoints on the X- axis are mapped to the vertices of the output space. So that the upper half-plane is transferred to a known potential in the course of the bent structure to be analyzed. The choice of the points is not always significant, it only affects the representation of the transformation. By forming the inverse transform and solving the complicated characteristics of the electric potential can be represented directly.