LectureNotes15U - Math 497C Nov 30, 2004 1 Curves and...

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 ....
View Full Document

This note was uploaded on 08/25/2011 for the course MATH 6456 taught by Professor Staff during the Spring '08 term at Georgia Institute of Technology.

Page1 / 5

LectureNotes15U - Math 497C Nov 30, 2004 1 Curves and...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online