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Pedal curve
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===From parametric equations=== [[Image:Contrapedal.gif|500px|right|thumb|Contrapedal of the same ellipse]] [[Image:PedalCurve3.gif|500px|right|thumb|Pedal of the evolute of the ellipse : same as the contrapedal of the original ellipse]] Let <math>\vec{v} = P - R</math> be the vector for ''R'' to ''P'' and write :<math>\vec{v} = \vec{v}_{\parallel}+\vec{v}_\perp</math>, the [[tangential and normal components]] of <math>\vec{v}</math> with respect to the curve. Then <math>\vec{v}_{\parallel}</math> is the vector from ''R'' to ''X'' from which the position of ''X'' can be computed. Specifically, if ''c'' is a [[parametric curve|parametrization]] of the curve then :<math>t\mapsto c(t)+{ c'(t) \cdot (P-c(t))\over|c'(t)|^2} c'(t)</math> parametrises the pedal curve (disregarding points where ''c' ''is zero or undefined). For a parametrically defined curve, its pedal curve with pedal point (0;0) is defined as :<math>X[x,y]=\frac{(xy'-yx')y'}{x'^2 + y'^2}</math> :<math>Y[x,y]=\frac{(yx'-xy')x'}{x'^2 + y'^2}.</math> The contrapedal curve is given by: :<math>t\mapsto P-{ c'(t) \cdot (P-c(t))\over|c'(t)|^2} c'(t)</math> With the same pedal point, the contrapedal curve is the pedal curve of the [[evolute]] of the given curve.
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