Automatic Generation of Riemann Surface Meshes

Abstract

Riemann surfaces naturally appear in the analysis of complex functions that are branched over the complex plane. However, they usually possess a complicated topology and are thus hard to understand. We present an algorithm for constructing Riemann surfaces as meshes in ${\mathbb R}^3$ from explicitly given branch points with corresponding branch indices. The constructed surfaces cover the complex plane by the canonical projection onto ${\mathbb R}^2$ and can therefore be considered as multivalued graphs over the plane – hence they provide a comprehensible visualization of the topological structure.

Complex functions are elegantly visualized using domain coloring on a subset of ${\mathbb C}$. By applying domain coloring to the automatically constructed Riemann surface models, we generalize this approach to deal with functions which cannot be entirely visualized in the complex plane.