Convergence Analysis of Discrete Conformal Transformation

TL;DR

This paper proves that discrete conformal mappings on triangulated meshes converge to continuous conformal maps as the mesh becomes finer, resolving an open problem in geometric analysis.

Contribution

It provides a rigorous convergence proof for discrete conformal transformations under weak triangulation conditions, advancing understanding in discrete differential geometry.

Findings

Discrete conformal energy minimizers converge to continuous conformal maps

Convergence holds under weak triangulation conditions

Error analysis confirms the theoretical convergence result

Abstract

Continuous conformal transformation minimizes the conformal energy. The convergence of minimizing discrete conformal energy when the discrete mesh size tends to zero is an open problem. This paper addresses this problem via a careful error analysis of the discrete conformal energy. Under a weak condition on triangulation, the discrete function minimizing the discrete conformal energy converges to the continuous conformal mapping as the mesh size tends to zero.