Ellipse Negative Pedal Curve

Author

Eric W. Weisstein
March 16, 2008

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

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

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

Equations

np=FullSimplify[NegativePedalCurve[{a Cos[t],b Sin[t]},{x,y},t]]

(0, 0)

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

eq=Simplify[%/.Cos[t]^2->(1-Sin[t]^2)/.Cos[t]^3->Cos[t](1-Sin[t]^2)]

FullSimplify[eq/.b^2->a^2-c^2]

Factor[FullSimplify[eq/.b^2->a^2-c^2]/.c->a e/.b->a Sqrt[1-e^2]]

eq/.t->0

movie=With[{a=1},Table[ParametricPlot[{{Cos[t],b Sin[t]},{((Cos[t] ((a^2)+((a^2)-(b^2)) ((Sin[t])^2)))/a),((Sin[t] (-(a^2)+2 (b^2)+((a^2)-(b^2)) ((Sin[t])^2)))/b)}},{t,0,2Pi},
PlotStyle->{Black,Red},
AspectRatio->Automatic,
Epilog->{Red,PointSize[.02],Point[{0,0}]},
PlotRange->{{-1,1},{-1,1}}1.1,
PlotLabel->TraditionalForm["\!\(\*
StyleBox[\"b\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\"/\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\"a\",\nFontSlant->\"Italic\"]\) ="<>ToString[PaddedForm[b,{3,2}]]]
],{b,1,0.0001,-.02}]
]

Show[GraphicsArray[Partition[Take[movie,{1,Length[movie],5}],5,5,{1,1},{}]]]

1./Sqrt[2]

<<GIF`;

Export[gifs<>"ellnegp0.gif",movie]//Timing

(c, 0)

eq=FullSimplify[np/.{x->Sqrt[a^2-b^2],y->0}]

FullSimplify[eq/.(Sqrt[(a-b) (a+b)])->c]

FullSimplify[%/.b^2->a^2-c^2]

b/a=1

FullSimplify[{a Cos[t]-(Sqrt[(a-b) (a+b)]) ((Sin[t])^2),(((-(a^2)+2 (b^2)+a (Sqrt[(a-b) (a+b)]) Cos[t]) Sin[t])/b)}/.{b->a}]

(b/a)=(1/(Sqrt[2]))

fish curve

Images

moviec=With[{a=1},Table[ParametricPlot[{{Cos[t],b Sin[t]},{a Cos[t]-(Sqrt[(a-b) (a+b)]) ((Sin[t])^2),(((-(a^2)+2 (b^2)+a (Sqrt[(a-b) (a+b)]) Cos[t]) Sin[t])/b)}},{t,0,2Pi},
PlotStyle->{Black,Red},
AspectRatio->Automatic,
Epilog->{Red,PointSize[.02],Point[{Sqrt[a^2-b^2],0}]},
PlotRange->{{-1,1},{-1,1}}1.1,
PlotLabel->TraditionalForm["\!\(\*
StyleBox[\"b\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\"/\",\nFontSlant->\"Italic\"]\)\!\(\*
StyleBox[\"a\",\nFontSlant->\"Italic\"]\) ="<>ToString[PaddedForm[b,{3,2}]]]
],{b,1,0.0001,-.02}]
]

Show[GraphicsArray[Partition[Take[moviec,{1,Length[movie],5}],5,5,{1,1},{}]],ImageSize->450]

Export[gifs<>"ellnegpc.gif",moviec]//Timing