In[]:=
OrthopticCurveEllipse::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; AI Assistant; Personal.";
Development:
Obtaining the cartesian equation from the orthoptic curve to the ellipse:
In[]:=
2
x
2
a
+
2
y
2
b
==1;​​y-
y
0
==m(x-
x
0
);​​y==
y
0
+mx-m
x
0
;​​
2
x
2
a
+
2
(
y
0
+mx-m
x
0
)
2
b
==1;​​Expand[
2
b
2
x
+
2
a
2
(
y
0
+mx-m
x
0
)
==
2
a
2
b
]
Out[]=
9
2
x
+16
2
m
2
x
-32
2
m
x
x
0
+16
2
m
2
x
0
+32mx
y
0
-32m
x
0
y
0
+16
2
y
0
144
In[]:=
2
b
2
x
+
2
a
2
m
2
x
-2
2
a
2
m
x
x
0
+
2
a
2
m
2
x
0
+2
2
a
mx
y
0
-2
2
a
m
x
0
y
0
+
2
a
2
y
0

2
a
2
b
;
In[]:=
(
2
b
+
2
a
2
m
)
2
x
+-2
2
a
2
m
x
0
+2
2
a
m
y
0
x+
2
a
2
m
2
x
0
-2
2
a
m
x
0
y
0
+
2
a
2
y
0
-
2
a
2
b
==0;
Due to the condition of tangency, the discriminant must be zero:
In[]:=
SimplifyExpand
2
-2
2
a
2
m
x
0
+2
2
a
m
y
0

-4(
2
b
+
2
a
2
m
)
2
a
2
m
2
x
0
-2
2
a
m
x
0
y
0
+
2
a
2
y
0
-
2
a
2
b
==0
Out[]=
2
m
2
x
0
+
2
y
0
9+16
2
m
+2m
x
0
y
0
In[]:=
ab
2
b
+
2
a
2
m
-
2
m
2
x
0
+2m
x
0
y
0
-
2
y
0
0;
In[]:=

2
a
-
2
x
0

2
m
+(2
x
0
y
0
)m-
2
y
0
+
2
b
==0;
By Vieta’s theorem:
In[]:=
m
1
m
2
==
c
a
;
Due to the perpendicularity condition:
In[]:=
-
2
y
0
+
2
b
2
a
-
2
x
0
==-1;​​
2
x
0
+
2
y
0
==
2
a
+
2
b
;​​
2
x
+
2
y
==
2
a
+
2
b
;
Animation:
In[]:=
Manipulatea=4;b=3;​​Module{p,slopes,m1,m2,t1Line,t2Line},​​​​p=
2
a
+
2
b
Cos[t1],
2
a
+
2
b
Sin[t1];​​​​slopes=m/.Solve[​​(
2
a
-
2
p[[1]]
)
2
m
+2p[[1]]p[[2]]m+(
2
b
-
2
p[[2]]
)==0,​​m​​]//Simplify;​​​​{m1,m2}=slopes;​​​​t1Line[x_]:=p[[2]]+m1(x-p[[1]]);​​t2Line[x_]:=p[[2]]+m2(x-p[[1]]);​​​​Show​​​​ParametricPlot[{aCos[θ],bSin[θ]},{θ,0,2π},PlotStyle->{Blue,Thick}],​​ParametricPlot
2
a
+
2
b
Cos[θ],
2
a
+
2
b
Sin[θ],{θ,0,t1},​​ColorFunction->Function[{x,y,u},ColorData["BlueGreenYellow"][u]],​​ColorFunctionScaling->True​​,​​Plot[t1Line[x],{x,-6,6},PlotStyle->Black],​​Plot[t2Line[x],{x,-6,6},PlotStyle->Black],​​Graphics[{​​Black,PointSize[0.01],Point[p],​​Text[Style["P",14,Bold],p,{0,-1.2}]​​}]​​,​​AspectRatio->Automatic,​​Axes->True,​​AxesOrigin->{0,0},​​AxesLabel->{x,y},​​PlotRange->6,​​ImageSize->900​​​​,​​{{t1,0.00002,"Value t1"},0.000001,2π},SaveDefinitions->True​​
Out[]=
​
Value t1