OrthopticCurveAstroid::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 astroid:
In[]:=
ζ[t_]:={a
3
Cos[t]
,a
3
Sin[t]
};​​x[t_]:=a
3
Cos[t]
;​​y[t_]:=a
3
Sin[t]
;​​
y'[t]
x'[t]
//FullSimplify
-Tan[t];
In[]:=
m[t_]:=-Tan[t];
Perpendicularity condition (be two points of the astroid
t
1
and
t
2
):
Tan[
t
1
]Tan[
t
2
]==-1;
This condition is only met if the difference of the angles is
π
2
:
t
2
-
t
1
==
π
2
;
t
2
==
t
1
+
π
2
;
In[]:=
​​Solvey-y[t]==m[t](x-x[t]),y-yt+
π
2
==mt+
π
2
x-xt+
π
2
,{x,y}//FullSimplify
{{xaCos[t]Sin[t](-Cos[t]+Sin[t]),yaCos[t]Sin[t](Cos[t]+Sin[t])}};
​​
ζ
O
[t_]:={aCos[t]Sin[t](-Cos[t]+Sin[t]),aCos[t]Sin[t](Cos[t]+Sin[t])};
Animation:
In[]:=
Manipulatea=3;​​ζ[t_]:={a
3
Cos[t]
,a
3
Sin[t]
};​​
ζ
O
[t_]:={aCos[t]Sin[t](-Cos[t]+Sin[t]),aCos[t]Sin[t](Cos[t]+Sin[t])};​​​​Show​​ParametricPlot[ζ[θ],{θ,0,2π},PlotStyle{Blue,Thick}],​​ParametricPlot[
ζ
O
[t],{t,0,t1},ColorFunction->"BlueGreenYellow"],​​GraphicsBlack,PointSize[0.007],​​Point[ζ[t1]],Text[Style["
P
1
",14,Bold],ζ[t1],{0,-1.2}],​​Pointζt1+
π
2
,TextStyle["
P
2
",14,Bold],ζt1+
π
2
,{0,-1.2},​​Point[
ζ
O
[t1]],Text[Style["
ζ
O
",14,Bold],
ζ
O
[t1],{0,-1.2}],​​InfiniteLine[{ζ[t1],
ζ
O
[t1]}],InfiniteLineζt1+
π
2
,
ζ
O
[t1],AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange4,ImageSize900,Style["Orthoptic curve of an astroid",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,2π,0.00001},ControlPlacementTop
Out[]=
​
Orthoptic curve of an astroid
Value (t1)