LectureNotes3U

LectureNotes3U - Math 497C Sep 17, 2004 1 Curves and...

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

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

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: Math 497C Sep 17, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 3 1.8 The general definition of curvature; Fox-Milnors Theorem Let : [ a, b ] R n be a curve and P = { t , . . . , t n } be a partition of [ a, b ], then (the approximation of) the total curvature of with respect to P is defined as total [ , P ] := n 1 X i =1 angle ( t i ) ( t i 1 ) , ( t i +1 ) ( t i ) , and the total curvature of is given by total [ ] := sup { [ , P ] | P P artition [ a, b ] } . Our main aim here is to prove the following observation due to Ralph Fox and John Milnor: Theorem 1 (Fox-Minor). If : [ a, b ] R n is a C 2 unit speed curve, then total [ ] = Z b a k ( t ) k dt. This theorem implies, by the mean value theorem for integrals, that for any t ( a, b ), ( t ) = lim 1 2 total h t + t i . The above formula may be taken as the definition of curvature for general (not necessarily C 2 ) curves. To prove the above theorem first we need to develop some basic spherical geometry. Let S n := { p R n +1 | k p k = 1 } . 1 Last revised: September 17, 2004 1 denote the n-dimensional unit sphere in R n +1 . Define a mapping from S n S n to R by dist S n ( p, q ) := angle( p, q ) . Exercise 2. Show that ( S n , dist S n ) is a metric space. The above metric has a simple geometric interpretation described as fol- lows. By a great circle C S n we mean the intersection of S n with a two dimensional plane which passes through the origin o of R n +1 . For any pair of points p , q S 2 , there exists a plane passing through them and the origin., there exists a plane passing through them and the origin....
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

LectureNotes3U - Math 497C Sep 17, 2004 1 Curves and...

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

View Full Document
Ask a homework question - tutors are online