The Discrete Theorema Egregium

IF 0.4 4区数学 Q4 MATHEMATICS

American Mathematical Monthly Pub Date : 2023-10-20 DOI:10.1080/00029890.2023.2263299

Thomas F. Banchoff, Felix Günther

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引用次数: 0

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

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离散定理

1827年，高斯证明了高斯曲率实际上是一个本征量，这意味着它可以通过表面内部的测量来计算。在此之前，曲面的曲率只能从外部计算，这意味着需要一个环境空间。高斯将这一非凡的发现命名为“惊人定理”。本文讨论了该定理在多面体曲面上的离散化形式。给出了离散高斯曲率的一般外在定义和内在定义是等价的一个初等证明。作者感谢两位匿名审稿人和编委会的宝贵意见。这项研究是在第二作者在波恩的马克斯普朗克数学研究所逗留期间开始的。该项目由德国研究基金会(Deutsche Forschungsgemeinschaft DFG)通过“几何和动力学离散化”合作研究中心TRR 109和德国卓越战略-柏林数学研究中心MATH+ (ec -2046/1，项目ID: 390685689)资助。作者简介:thomas F. BANCHOFF，布朗大学数学名誉教授，1967年至2015年任教于布朗大学。他于1964年在加州大学伯克利分校获得博士学位，并于1999年至2000年担任MAA主席。布朗大学数学系，邮编:151 Thayer Street, Providence RI 02912Thomas_Banchoff@brown.eduFelix g ntherfelix GÜNTHER, 2014年获得柏林理工大学Universität数学博士学位。在伊维特河畔布尔斯高等科学研究所、剑桥艾萨克·牛顿数学科学研究所、维也纳欧文Schrödinger国际数学与物理研究所、波恩马克斯·普朗克数学研究所和日内瓦大学担任博士后后，他于2018年回到Universität柏林工业大学。主要研究方向为离散微分几何和离散复分析。他还热衷于科学传播。柏林理工大学Universität，德国数学研究所8-3,Straße des 17。Juni 136, 10623柏林，Germanyfguenth@math.tu-berlin.de

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来源期刊

American Mathematical Monthly Mathematics-General Mathematics

CiteScore

0.80

自引率

20.00%

发文量

127

审稿时长

6-12 weeks

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