The Discrete Theorema Egregium

In 1827, Gauss proved that Gaussian curvature is actually an intrinsic quantity, meaning that it can be calculated just from measurements within the surface. Before, curvature of surfaces could only be computed extrinsically, meaning that an ambient space is needed. Gauss named this remarkable finding Theorema Egregium. In this paper, we discuss a discrete version of this theorem for polyhedral surfaces. We give an elementary proof that the common extrinsic and intrinsic definitions of discrete Gaussian curvature are equivalent.

Automated summary

This paper discusses a discrete version of the Theorema Egregium for polyhedral surfaces and gives an elementary proof of the equivalence between intrinsic and extrinsic definitions of discrete Gaussian curvature.