Parabola Negative Pedal Curve

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Equations

With[{a=1},
ParametricPlot[{a t^2,2 a t},{t,-2,2}]
]

np=FullSimplify[NegativePedalCurve[{a t^2,2 a t},{x,y},t]]

(0, 0)

FullSimplify[np/.{x->0,y->0}]

With[{a=1},ParametricPlot[{a (4+3 (t^2)),-a (t^3)},{t,-2,2}]]

(x, 0)

FullSimplify[np/.y->0]

(a, 0)

FullSimplify[np/.{x->a,y->0}]

With[{a=1},ParametricPlot[{{a t^2,2a t},{3 a (t^2),-a t (-3+(t^2))}},{t,-5,5},
PlotStyle->{Black,Red},
Ticks->None,
Epilog->{Red,PointSize[.03],Point[{a,0}]},
PlotRange->{{-.1,15},{-8,8}},
AspectRatio->Automatic]]

(0, y)

FullSimplify[np/.x->0]

Animation

Table[x,{x,-.5,3,.1}]

movie=With[{a=1},Table[ParametricPlot[{{a t^2,2a t},{((a (t^2) (a (4+3 (t^2))-x))/(a (t^2)+x)),-((t ((a^2) (t^4)-2 a (2+(t^2)) x+(x^2)))/(a (t^2)+x))}},{t,-4,4},
PlotStyle->{Black,Red},
AspectRatio->Automatic,
Epilog->{Red,PointSize[.02],Point[{x,0}]},
PlotRange->{{-2,15},{-8,8}},
Ticks->None,
PlotLabel->TraditionalForm["\!\(\*
StyleBox[\"x\",\nFontSlant->\"Italic\"]\) ="<>ToString[PaddedForm[Chop[x],{3,2}]]]
],{x,-2,4,.2}]
]

<<Utilities`GIF`;

Export[gifs<>"paranegp.gif",movie]//Timing

Show[GraphicsArray[Partition[Take[movie/.PointSize[x_]:>PointSize[.06],{1,Length[movie],5}],4,4,{1,1},{}]],ImageSize->450]