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

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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....
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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.

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

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