GPU-based visualization of domain-coloured algebraic Riemann surfaces

TL;DR

This paper presents an efficient GPU-based algorithm for visualizing domain-coloured Riemann surfaces of algebraic curves, enabling interactive exploration of complex mathematical objects with accurate topology and structure.

Contribution

It introduces a GPU-accelerated method for visualizing Riemann surfaces that faithfully reproduces their topology and holomorphic structure, implemented in OpenGL and WebGL.

Findings

Interactive visualization of complex algebraic curves achieved

Algorithm faithfully reproduces topology and structure

Visualization reveals features not obvious from equations

Abstract

We examine an algorithm for the visualization of domain-coloured Riemann surfaces of plane algebraic curves. The approach faithfully reproduces the topology and the holomorphic structure of the Riemann surface. We discuss how the algorithm can be implemented efficiently in OpenGL with geometry shaders, and (less efficiently) even in WebGL with multiple render targets and floating point textures. While the generation of the surface takes noticeable time in both implementations, the visualization of a cached Riemann surface mesh is possible with interactive performance. This allows us to visually explore otherwise almost unimaginable mathematical objects. As examples, we look at the complex square root and the folium of Descartes. For the folium of Descartes, the visualization reveals features of the algebraic curve which are not obvious from its equation.