The video “Geodesics and Waves”

Abstract

ABSTRACT The video Geodesics and Waves introduces the concepts ofstraightest geodesics and geodesic flow on polyhedral surfaces. Itis the third in a series of videos presenting research results fromthe area of mathematics and visualization produced at thedepartment of Experimental Mathematics and Visualization at theSonderforschungsbereich 288 on Differential Geometry andQuantum Physics at the Technical University of Berlin (see[1][2]). Keywords Geodesics, Geodesic Flow, Isometric Texture Maps, PolyhedralSurfaces 1. INTRODUCTION Geodesic curves are a fundamental concept in geometry thatgeneralize the idea of straight lines in the plain to curves in curvedsurfaces and arbitrary manifolds. Smooth geodesics are locallyshortest curves and have vanishing geodesic curvature. Onpolyhedral surfaces these two properties are no longer equivalent.Therefore we introduce the notion of discrete geodesic curvaturefor curves and use it for the definition of straightest geodesics onpolyhedral surfaces. In contrast to the widely used shortestgeodesics, which fail to pass through positively curved surfacevertices, straightest geodesics uniquely solve the initial valueproblem for geodesics.The unique solvability of the geodesic initial value problem bystraightest geodesics directly leads to a definition of paralleltranslation of vectors along curves. An immediate application isthe definition of a geodesic Runge-Kutta method for theintegration of vector fields on polyhedral surfaces.