The Discrete Theorema Egregium

Abstract

AbstractIn 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. AcknowledgmentThe authors thank the two anonymous reviewers and the editorial board for their valuable remarks. The research was initiated during the second author’s stay at the Max Planck Institute for Mathematics in Bonn. It was funded by the Deutsche Forschungsgemeinschaft DFG through the Collaborative Research Center TRR 109 “Discretization in Geometry and Dynamics” and under Germany’s Excellence Strategy – The Berlin Mathematics Research Center MATH+ (EXC-2046/1, project ID: 390685689).Additional informationNotes on contributorsThomas F. BanchoffTHOMAS BANCHOFF is professor emeritus of mathematics at Brown University, where he taught from 1967 to 2015. He received his Ph.D. from the University of California, Berkeley, in 1964 and served as president of the MAA from 1999 to 2000.Department of Mathematics, Brown University, Box 1917, 151 Thayer Street, Providence RI 02912Thomas_Banchoff@brown.eduFelix GüntherFELIX GÜNTHER received his Ph.D. in mathematics from Technische Universität Berlin in 2014. After holding postdoctoral positions at the Institut des Hautes Etudes Scientifiques in Bures-sur-Yvette, the Isaac Newton Institute for Mathematical Sciences in Cambridge, the Erwin Schrödinger International Institute for Mathematics and Physics in Vienna, the Max Planck Institute for Mathematics in Bonn, and the University of Geneva, he came back to Technische Universität Berlin in 2018. His research interests include discrete differential geometry and discrete complex analysis. He also has a passion for science communication.Technische Universität Berlin, Institut für Mathematik MA 8-3, Straße des 17. Juni 136, 10623 Berlin, Germanyfguenth@math.tu-berlin.de