Circle Pedal Curve

Author

Eric W. Weisstein
June 27, 2013

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

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

©2013 Wolfram Research, Inc. except for portions noted otherwise

Equation

Parametric

circle=a{Cos[t],Sin[t]};

eqn=FullSimplify[PedalCurve[circle,{x0,y0},t],0<t<2π]

eqn2=Together[TrigExpand[eqn]/.{Sin[t]->sin,Cos[t]->cos}]

eqns=2{x,y}-Numerator[eqn2]

gb=Factor/@GroebnerBasis[Append[eqns,sin^2+cos^2-1],{x,y},{sin,cos}]//FullSimplify

Length[gb]

AlgebraicDegree[gb[[1]],{x,y}]

Properties

par={x0 ((Sin[t])^2)+Cos[t] (a-y0 Sin[t]),(1/2) (y0+y0 Cos[2 t]+2 a Sin[t]-x0 Sin[2 t])};

ArcLength

(s=ArcLength[par,{t,0,2π},Assumptions->a>0&&(x0|y0)∈Reals&&x0^2+y0^2<a^2])//Timing

(s=ArcLength[par,{t,0,2π},Assumptions->a>0&&(x0|y0)∈Reals&&x0^2+y0^2==a^2])//Timing

(s=ArcLength[par,{t,0,2π},Assumptions->a>0&&(x0|y0)∈Reals&&x0^2+y0^2>a^2])//Timing

ArcLengthFunction

ArcLength[par,t,Assumptions->a>0&&(x0|y0)∈Reals&&x0^2+y0^2<a^2&&t>0]//Timing

ArcLength[par,t,Assumptions->a>0&&(x0|y0)∈Reals&&x0^2+y0^2==a^2&&t>0]//Timing

ArcLength[par,t,Assumptions->a>0&&(x0|y0)∈Reals&&x0^2+y0^2>a^2&&t>0]//Timing

Area

Assuming[(x0|y0)∈Reals&&a>0,Area[par,{t,0,2π}]]//Timing

AreaInertiaTensor

Assuming[(x0|y0)∈Reals&&a>0,AreaInertiaTensor[par,{t,0,2π}]]//Timing

FullSimplify[Last[%]]

Centroid

Assuming[(x0|y0)∈Reals&&a>0,Centroid[par,{t,0,2π}]]//Timing

Curvature

FullSimplify[Curvature[par,t],a>0&&0<t<2π]

FullSimplify[Curvature[((((x^2)-x x0+y (y-y0)))^2)-(a^2) ((((x-x0))^2)+(((y-y0))^2)),{x,y}],a>0]

Slope

Slope[{x0 Sin[t]^2+Cos[t] (a-y0 Sin[t]),1/2 (y0+y0 Cos[2 t]+2 a Sin[t]-x0 Sin[2 t])},t]//FullSimplify

Slope[(x^2-x x0+y (y-y0))^2==a^2 ((x-x0)^2+(y-y0)^2),{x,y}]//FullSimplify

Slope[(x^2-x x0+y (y-y0))^2==a^2 ((x-x0)^2+(y-y0)^2),{x,y}]//FullSimplify

SpecialAffineCurvature

FullSimplify[SpecialAffineCurvature[par,t],a>0&&(x0|y0)∈Reals]

TangentialAngle

FullSimplify[TangentialAngle[par,t],a>0]

FullSimplify[TangentialAngle[par,t],a>0&&0<t<2π]

Plots

At Center

FullSimplify[PedalCurve[circle,{0,0},t]]

On Circumference

FullSimplify[PedalCurve[circle,{1,0},t]]

ParametricPlot[{Cos[t]+((Sin[t])^2),-(-1+Cos[t]) Sin[t]},{t,0,2π},Ticks->None,AspectRatio->Automatic,PlotStyle->Red]

PedalCurvePlot[{Cos[t],Sin[t]},{1,0},{t,0,2Pi},{0,3,.2}]

Off Circumference

PedalCurvePlot[circle,{1.5,0},{t,0,2Pi},{0,3,.2}]

Array of Curves

PedalCurve[{Cos[t],Sin[t]},{1,1},t]

Show[GraphicsArray[{PedalCurvePlot[{Cos[t],Sin[t]},{.5,0},{t,0,2Pi},{0,3,.2},DisplayFunction->Identity],PedalCurvePlot[{Cos[t],Sin[t]},{.7,0},{t,0,2Pi},{0,3,.2},DisplayFunction->Identity],PedalCurvePlot[{Cos[t],Sin[t]},{1,0},{t,0,2Pi},{0,3,.2},DisplayFunction->Identity],PedalCurvePlot[{Cos[t],Sin[t]},{1.5,0},{t,0,2Pi},{0,3,.2},DisplayFunction->Identity]}],
DisplayFunction->$DisplayFunction]

V6

p[{x0_,y0_},t_,a_:1]:={x0 ((Sin[t])^2)+Cos[t] (a-y0 Sin[t]),(1/2) (y0+y0 Cos[2 t]+2 a Sin[t]-x0 Sin[2 t])}

GraphicsRow[{
ParametricPlot[p[{1,0},t],{t,0,2π}],
ParametricPlot[p[{1.2,0},t],{t,0,2π}],
ParametricPlot[p[{2,0},t],{t,0,2π}],
ParametricPlot[p[{1,1},t],{t,0,2π}]
}]

GraphicsRow[{
ParametricPlot[p[{1,0},t],{t,0,π}],
ParametricPlot[p[{1.2,0},t],{t,0,π}],
ParametricPlot[p[{2,0},t],{t,0,π}],
ParametricPlot[p[{1,1},t],{t,0,π}]
}]

Vector Properties

parm={x0 Sin[t]^2+Cos[t] (a-y0 Sin[t]),1/2 (y0+y0 Cos[2 t]+2 a Sin[t]-x0 Sin[2 t])};
assum=a>0&&(x0|y0)∈Reals&&0<t<2π;
xform=Abs[x_]^(n_/;Mod[n,2]==0):>x^n;

VectorLength

Assuming[assum,FullSimplify[Norm[parm]]]/.xform

NormalVector

Assuming[assum,FullSimplify[NormalVector[parm,t]]]/.xform

TangentVector

Assuming[assum,FullSimplify[TangentVector[parm,t]]]/.xform