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OrthopticCurveCardioid::usage="It is the geometric locus of the intersection point of two tangent lines to a curve and orthogonal to each other. References: mathecurve.com: Orthoptic curve; Personal.";
Development:
Obtaining the parametric equations from the orthoptic curve to the cardioid:
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ζ[t_]:={aCos[t](1+Cos[t]),aSin[t](1+Cos[t])};x[t_]:=aCos[t](1+Cos[t]);y[t_]:=aSin[t](1+Cos[t]);//FullSimplify
y'[t]
x'[t]
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-Cot
3t
2
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-Cot;
3t
2
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m[t_]:=-Cot;
3t
2
Perpendicularity condition (be two points of the cycloid and):
t
1
t
2
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CotCot==-1;
3
t
1
2
3
t
2
2
This condition is only met if the difference of the angles is :
π
2
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3
t
2
2
3
t
1
2
π
2
t
2
t
1
π
3
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Solvey-y[t]==m[t](x-x[t]),y-yt+==mt+x-xt+,{x,y}//FullSimplify
π
3
π
3
π
3
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x(4+9Cos[t]+3Cos[2t]-3CscSecCos(π-6t)+Cos+2t+Sin[t]+Sin[2t]Sin[3t]
3
8
3
(Sin[t]+Sin[2t])),y9
8
3t
2
3t
2
1
6
π
6
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xa2+3Cos[t]+3Cos[2t]+3Sin(π-6t)-3Sin+2t,ya[t_]:=a2+3Cos[t]+3Cos[2t]+3Sin(π-6t)-3Sin+2t,a
1
4
1
6
π
6
3
8
3
Cos[t]+3
Cos[2t]+3Sin[t]+Sin[2t];ζ
O
1
4
1
6
π
6
3
8
3
Cos[t]+3
Cos[2t]+3Sin[t]+Sin[2t];Animation:
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Manipulatea=3;ζ[t_]:={aCos[t](1+Cos[t]),aSin[t](1+Cos[t])};[t_]:=a2+3Cos[t]+3Cos[2t]+3Sin(π-6t)-3Sin+2t,a[t],{t,0,t1},ColorFunction->"BlueGreenYellow"],GraphicsBlack,PointSize[0.007],Point[ζ[t1]],Text[Style["",14,Bold],ζ[t1],{0,-1.2}],Pointζt1+,TextStyle["",14,Bold],ζt1+,{0,-1.2},Point[[t1]],Text[Style["",14,Bold],[t1],{0,-1.2}],InfiniteLine[{ζ[t1],[t1]}],InfiniteLineζt1+,[t1],AspectRatioAutomatic,AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange{{-2,8},{-6,6}},ImageSize900,Style["Orthoptic curve of a cardioid",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,2π,0.00001},ControlPlacementTop,SaveDefinitions->True
ζ
O
1
4
1
6
π
6
3
8
3
Cos[t]+3
Cos[2t]+3Sin[t]+Sin[2t];ShowParametricPlot[ζ[θ],{θ,0,2π},PlotStyle{Blue,Thick}],ParametricPlot[ζ
O
P
1
π
3
P
2
π
3
ζ
O
ζ
O
ζ
O
ζ
O
π
3
ζ
O
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