In[]:=
OrthopticCurveAstroid::usage="It is the geometric locus of the intersection point of two tangent lines to a curve and orthogonal to each other. References: mathcurve.com: Orthoptic curve; Personal.";
Development:
Obtaining the parametric equations from the orthoptic curve to the cross curve:
In[]:=
ζ[t_]:=
a
Sin[t]
,
a
Cos[t]
;​​x[t_]:=
a
Sin[t]
;​​y[t_]:=
a
Cos[t]
;​​
y'[t]
x'[t]
//FullSimplify
Out[]=
-
3
Tan[t]
In[]:=
-
3
Tan[t]
;
In[]:=
m[t_]:=-
3
Tan[t]
;;
Perpendicularity condition (be two points of the astroid
t
1
and
t
2
):
In[]:=
3
Tan[
t
1
]
3
Tan[
t
2
]
==-1;
This condition is only met if the difference of the angles is
π
2
:
In[]:=
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
Out[]=
x
3(1+
3
Cot[t]
)Csc[t]
1-
2
Cot[t]
+
4
Cot[t]
,y
3(-1+
3
Cot[t]
)Csc[t]
1-
2
Cot[t]
+
4
Cot[t]

In[]:=
x
a(1+
3
Cot[t]
)Csc[t]
1-
2
Cot[t]
+
4
Cot[t]
,y
4a(Cos[t]-Sin[t])(2+Sin[2t])
5+3Cos[4t]
;
In[]:=
​​
ζ
O
[t_]:=
a(1+
3
Cot[t]
)Csc[t]
1-
2
Cot[t]
+
4
Cot[t]
,
4a(Cos[t]-Sin[t])(2+Sin[2t])
5+3Cos[4t]
;
Animation:
In[]:=
Manipulatea=3;​​ζ[t_]:=
a
Sin[t]
,
a
Cos[t]
;​​
ζ
O
[t_]:=
a(1+
3
Cot[t]
)Csc[t]
1-
2
Cot[t]
+
4
Cot[t]
,
4a(Cos[t]-Sin[t])(2+Sin[2t])
5+3Cos[4t]
;​​Show​​ParametricPlot[ζ[θ],{θ,0,2π},PlotStyle{Blue,Thick},PerformanceGoal->"Quality",PlotPoints->60],​​ParametricPlot[
ζ
O
[t],{t,0,t1},ColorFunction->"BlueGreenYellow",PerformanceGoal->"Quality",PlotPoints->60],​​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10,ImageSize900,Style["Orthoptic curve of a cross curve",Bold,Large],{{t1,0.00002,"Value (t1)"},0.00001,2π,0.00001},ControlPlacementTop,SaveDefinitions->True
Out[]=
​
Orthoptic curve of a cross curve
Value (t1)