PedalCurveCircumference::usage="
​References:https://e.math.cornell.edu>people;Personal.";
Development to obtain the parametric equations of the cardioid, which is the pedal curve of the circumference:
In[]:=
ζ[t_]:={Cos[t],Sin[t]};​​
ζ
P
=ζ[t]+
ζ'[t].({3,0}-ζ[t])
2
Norm[ζ'[t]]
ζ'[t]//FullSimplify
Out[]=
Cos[t]+
3
2
Sin[t]
2
Abs[Cos[t]]
+
2
Abs[Sin[t]]
,
(
2
Abs[Cos[t]]
+
2
Abs[Sin[t]]
-3Cos[t])Sin[t]
2
Abs[Cos[t]]
+
2
Abs[Sin[t]]

Cos[t]+
3
2
Sin[t]
2
Abs[Cos[t]]
+
2
Abs[Sin[t]]
,
(
2
Abs[Cos[t]]
+
2
Abs[Sin[t]]
-3Cos[t])Sin[t]
2
Abs[Cos[t]]
+
2
Abs[Sin[t]]
;​​
ζ
P
[t_]:={Cos[t]+3
2
Sin[t]
,(1-3Cos[t])Sin[t]};
Animation:
In[]:=
​​SwatchLegend[{Black,Blue,DarkGreen,Red},{"Circumference","Pedal curve (cardioid)","Tangent line to the circumference","Orthogonal line to the tangent line and the point {3, 0}"}]​​Manipulate[ζ[t_]:={Cos[t],Sin[t]};
ζ
P
[t_]:={Cos[t]+3
2
Sin[t]
,(1-3Cos[t])Sin[t]};​​Show[ParametricPlot[ζ[t],{t,0,2π},PlotStyle{Black,Thick}],​​Graphics[{Blue,PointSize[0.01],Point[
ζ
P
[k1]]}],Graphics[{Red,PointSize[0.01],Point[{3,0}]}],​​ParametricPlot[
ζ
P
[t],{t,-π,k1},PlotStyle{Blue,Thick},PerformanceGoal->"Quality"],​​Graphics[{Red,Thick,InfiniteLine[{{3,0},
ζ
P
[k1]}]}],Graphics[{DarkGreen,Thick,InfiniteLine[{ζ[k1],
ζ
P
[k1]}]}]​​,AxesLabel->(Style[#,15,Blue]&/@{"X","Y"}),AxesOrigin{0,0},AxesTrue,AspectRatioAutomatic,PlotRange{{-2,4},{-2.5,2.5}},ImageSize900​​],{k1,-π+0.00001,π,0.00001},SaveDefinitionsTrue,ControlPlacementTop]
Out[]=
Circumference
Pedal curve (cardioid)
Tangent line to the circumference
Orthogonal line to the tangent line and the point {3, 0}
Out[]=
​
k1