handout 13-4 - Math 200, Spring 2010 Handout 8 Section 13-4...

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Math 200, Spring 2010 Handout 8 Section 13-4 Curvature Definition: Curvature The curvature of C at a given point is a measure of how quickly the curve changes direction at that point. Specifically, curvature is the magnitude of the rate of change of the unit tangent vector with respect to arc length. Let ( ) s r be an arc length parametrization and ( ) t T the unit tangent vector. Then, the curvature of curve ( ) s r is ( ) d s ds κ = T Expressed in terms of parameter t (instead of arc length parameter s ), the curvature is ( ) ( ) ( ) t d d dt t ds d s dt t = = = T T T r Curvature is large where the unit tangent changes direction rapidly 1. Compute the curvature of a circle of radius R. Theorem: Formula for Curvature If ( ) t r is an regular parametrization, then the curvature at ( ) t r is ( ) ( ) ( ) ( ) 3 t t t t ′′ × = r r r . 2. Calculate the curvature ( ) t of the twisted cubic ( ) 2 3 , , t t t t = r .
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Formula for Curvature of a Plane Curve The curvature at the point
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This note was uploaded on 02/24/2011 for the course MATH 200 taught by Professor Jamesdcampbell during the Spring '10 term at Santa Barbara City.

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handout 13-4 - Math 200, Spring 2010 Handout 8 Section 13-4...

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