l6anno - Parametric Curve et ( s ) rOP = rOP ( s ) ds 2 =...

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Parametric Curve 2 '' 2 ' ' () 1( ) · · · OP OP OP OP OP OP OP OP OP tO P ds s ds s ds s ss d s s d s d s = = = = = = rr r er s is the distance along the curve Unit tangent vector t s e Not distinguishing between function and function value
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'' ' () () 1 ( · )( ) ( ) ( ) 0 ( ·· 2) 0 · tt t t ss s s = + = = ee e e ' · () 0 = Hence i.e., and are perpendicular Curvature indicates how much the path changes direction Principal unit normal ' t s e t s e ' t s e t s e ' t s e n s e ' ' |( ) | 1 nt t s s s = e
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Curvature Radius of curvature ' ' () | () | 11 () | () | t t ss s s s κ ρ = = = e e '' ' 1 () () |( ) | nt t t s s s == ee e e t s e t s s + Δ e t + Δ e t s e ' 0 1 () l im ( ( ) ) tt t s s s s Δ→ =+ Δ = Δ e Principal unit normal is perpendicular to and directed towards the concave side of the path ( ) s + Δ−
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Tangential and Normal Coordinates s=s(t) is the distance traveled () (() ) OP OP OP OP ts t s t = = = r r rr o Color change denotes different function (of time vs. arc length)
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Analogous to a parametric curve definition where t is the parameter, however P can reverse direction, although s increases monotonically (i.e. it never decreases so ) Multiple
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This note was uploaded on 01/26/2012 for the course TAM 212 taught by Professor Staff during the Spring '08 term at University of Illinois, Urbana Champaign.

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l6anno - Parametric Curve et ( s ) rOP = rOP ( s ) ds 2 =...

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