Parabola Pedal Curve

Author

Eric W. Weisstein
March 16, 2008

This notebook downloaded from http://mathworld.wolfram.com/notebooks/PlaneCurves/ParabolaPedalCurve.nb.

For more information, see Eric's MathWorld entry http://mathworld.wolfram.com/ParabolaPedalCurve.html.

©2008 Wolfram Research, Inc. except for portions noted otherwise

Equation

parabola={t^2,2t};

PedalCurve[a parabola,{(Subscript[x, 0]),(Subscript[y, 0])},t]//Simplify//Factor//TraditionalForm

Directrix

PedalCurvePlot[parabola,{-1,1},{t,-3,3},{0,Pi,.3},PlotRange->{-3,3}]

Foot of Directorix

PedalCurve[a parabola,{-1,0},t]//Simplify

Subtract@@Eliminate[Thread[{x,y}==%],t]

Solve[2 (x^2)+(x^3)-(-1-a) y2+x (1+y2)==0,y2]//Simplify

FullSimplify[-((x (((1+x))^2))/(1+a+x))/.x->x-1]

PedalCurvePlot[parabola,{-1,0},{t,-2,2},{0,Pi,.2}]

Reflection of Focus in Directorix

PedalCurvePlot[parabola,{-3,0},{t,-3,3},{0,Pi,.2},PlotRange->{-2,2}]

Vertex

PedalCurvePlot[parabola,{0,0},{t,-3,3},{0,Pi,.3}]

Focus

PedalCurvePlot[parabola,{1,0},{t,-2,2},{0,4,.4},PlotRange->{{-1,4},{-4,4}}]

Axis of Symmetry

PedalCurvePlot[parabola,{4,0},{t,-5,5},{0,4,.4},PlotRange->{{-1,4},{-5,5}}]

xy=FullSimplify[PedalCurve[{a+1-b t^2,2b t},{0,0},t]]

Expand[Subtract@@Eliminate[Thread[{x,y}==xy],t]]

Expand[(x-1)(x^2+y^2)-a x^2]