In[]:=
GeometricLocusAgnesi::usage="The line r rotates with center in O cutting to the circumference at point A and to the line S at point B. Find the geometric locus of the points obtained as an intersection of a line parallel to the X axis that passes through A with another line parallel to the axis Y that passes through B in all the positions of r. The diameter of the circumference is a (see the figure). References: Internet; Personal.​
";
Development:
Point A:
In[]:=
A[θ_]:=
a
2
Cos[θ],
a
2
Sin[θ]+
a
2
;
Point B:
Let the line y = m x be the one that passes through A, B and O.
In[]:=
Solve[{y==mx,y==a},{x,y}]
Out[]=
x
4
m
,y4
In[]:=
x
a
m
,ya;​​m=
a
2
Sin[θ]+
a
2
a
2
Cos[θ]
==
Sin[θ]+1
Cos[θ]
;​​B[θ_]:=
a
m
,a;​​B[θ_]:=
aCos[θ]
Sin[θ]+1
,a;
Parametric equations of the geometric locus (the geometric locus of the points obtained as an intersection of a line parallel to the X axis that passes through A with another line parallel to the axis Y that passes through B, that is
{
x
B
,
y
A
}
):
In[]:=
P
AB
={
x
B
,
y
A
};​​ζ[θ_]:=
aCos[θ]
Sin[θ]+1
,
a
2
Sin[θ]+
a
2
;
Cartesian equation:
In[]:=
Eliminatex==
aCos[θ]
Sin[θ]+1
,y==
a
2
Sin[θ]+
a
2
,θ
Out[]=
(16+
2
x
)y64
In[]:=
(
2
a
+
2
x
)y
3
a
&&
2
x
(-a+2y)
a
2
2
a
-
2
x
-2ay;
In[]:=
(
2
a
+
2
x
)y
3
a
;
It’s known as Agnesi’s witch
Animation:
In[]:=
Manipulatea=4;ζ[θ_]:=
aCos[θ]
Sin[θ]+1
,
a
2
Sin[θ]+
a
2
;A[θ_]:=
a
2
Cos[θ],
a
2
Sin[θ]+
a
2
;B[θ_]:=
aCos[θ]
Sin[θ]+1
,a;
P
x
[θ_]:=
aCos[θ]
Sin[θ]+1
,0;
P
y
[θ_]:=0,
a
2
Sin[θ]+
a
2
;O1={0,0};​​Line1[x_]:={x,a};​​Show​​ParametricPlot[ζ[θ],{θ,0,t},ColorFunction->"BlueGreenYellow"],​​ParametricPlot[Callout[Line1[x],"S",{Scaled[0.75],Below},LabelStyle{14,Bold,Black}],{x,-2π,2π},PlotStyle->{Black,Thick}],​​GraphicsBlack,PointSize[0.01],​​Point[O1],Text[Style["O",14,Bold],O1,{0,-1.2}],​​Point[A[t]],Text[Style["A",14,Bold],A[t],{0,-1.2}],​​Point[B[t]],Text[Style["B",14,Bold],B[t],{0,-1.2}],​​Point[ζ[t]],Text[Style["
P
AB
",14,Bold],ζ[t],{0,-1.2}],Thick,InfiniteLine[{A[t],B[t]}],{Dashed,InfiniteLine[{
P
x
[t],B[t]}],InfiniteLine[{
P
y
[t],A[t]}]},​​Blue,Circle0,
a
2
,
a
2
,AspectRatioAutomatic,​​AxesTrue,AxesOrigin{0,0},AxesLabel{x,y},PlotRange{{-2π,2π},{-1,2π}},ImageSize700,Style["Agnesi's witch",Bold,Large],{{t,0.00002,"Value (t)"},0.000001,2π,0.00001},AutorunSequencing->All,ControlPlacementTop,SaveDefinitions->True
Out[]=
​
Agnesi's witch
Value (t)