Discrete Differential Geometry: an Applied Introduction

domain M parameterization immersed surface Remember that, like the cartographers of yore, our goal is to parameterize a given surface over the plane. In particular, we want to find a map that preserves angles. How can we express this condition more explicitly? Well, we know how to express 90-degree rotations on the surface, using the complex structure J . And we know how to express 90-degree rotations in the plane, using the imaginary unit i. Therefore, an angle-preserving or conformal map z : M→ C must satisfy the Cauchy-Riemann equation