4. Constructing a Point on a Cassini Oval
4. Constructing a Point on a Cassini Oval
This Demonstration shows another construction of Cassini's oval. Start with the hyperbola with equation -=1 of eccentricity , . Select any point (u,v) on . Let be the opposite point of and a point on different from and . The tangents on at and are parallel and meet the tangent at at points and , respectively. Then =-+.
H
2
x
2
a
2
y
2
b
ϵ=+
2
a
2
b
a
a,b>0
D
1
H
D
2
D
1
D
H
D
1
D
2
H
D
1
D
2
D
E
1
E
2
D
1
E
1
D
2
E
2
2
a
2
ϵ
2
b
Draw a circle with center and radius and a circle with center and radius ; suppose these meet in points and . But then . So is a point on a Cassini oval with foci and . The same is true for the point . It can be shown that the foci and are also on the oval.
D
1
D
1
E
1
D
2
D
2
E
2
T
U
TT=-+
D
1
D
2
2
a
2
ϵ
2
b
T
D
1
D
2
U
F
1
F
2