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Unformatted text preview: Math 497C Nov 30, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 15 2.13 The Geodesic Curvature Let : I M be a unit speed curve lying on a surface M R 3 . Then the absolute geodesic curvature of is defined as | g | := ( 00 ) > = 00- 00 ,n ( ) n ( ) , where n is a local Gauss map of M in a neighborhood of ( t ). In particular note that if M = R 2 , then | g | = , i.e., absolute geodesic curvature of a curve on a surface is a gneralization of the curvature of curves in the plane. Exercise 1. Show that the absolute geodesic curvature of great circles in a sphere and helices on a cylinder are everywhere zero. Similarly, the (signed) geodesic curvature generalizes the notion of the signed curvature of planar curves and may be defined as follows. We say that a surface M R 3 is orientable provided that there exists a (global) Gauss map n : M S 2 , i.e., a continuous mapping which satisfies n ( p ) T p M , for all p M . Note that if n is a global Gauss map, then so is- n . In particular, any orientable surface admits precisely two choices for its global Gauss map. Once we choose a Gauss map n for an orientable surface, then M is said to be oriented . If M is an oriented surface (with global Gauss map n ), then, for every p M , we define a mapping J : T p M T p M by JV := n V. Exercise 2. Show that if M = R 2 , and n = (0 , , 1), then J is clockwise rotation about the origin by / 2. 1 Last revised: December 6, 2007 1 Then the geodesic curvature of a unit speed curve : I M is given by g := 00 ,J ....
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