The Schwarz-Christoffel transformation

A conformal map $f: \mathbb{C} \rightarrow \mathbb{C}$ is an angle-preserving transformation of points located in the complex plane. Conformal maps are particularly useful for solving electrostatics problems governed by Laplace's equation since solutions $w$ are harmonic and conformal maps preserve this property. Hence, if an electrostatics boundary value problem can be solved in one geometry, it can be mapped onto a different boundary geometry. The Schwarz-Christoffel transformation implements the mapping from a circular boundary to an arbitrary polygon.

This library implements

$$w = f(z) = c \int_0^z \prod_{k=1}^N \left(1-\frac{\tilde{z}}{z_k}\right)^{-\beta_k},{\rm d}\tilde{z} \ , $$

where $c$ is a constant for scaling and rotating, and $z_k \in \mathbb{C}$ are so-called prevertices which map to the vertices $w_k = f(z_k)$ of the polygon. Finding the prevertices to a given polygon is called the Schwarz-Christoffel parameter problem and is also implemented.

Vertices at infinity are also supported, enabling open boundary geometries.