3. Constructing a Point on a Cassini Oval

ellipse

a

1

b

0.6

point on ellipse

0.99

another point on ellipse

2.7

show Cassini oval

This Demonstration shows another ruler-and-compass construction of a point on a Cassini oval.

An ellipse

ℰ

is given with the equation

2

x

2

a

+

2

y

2

b

=1

and eccentricity

ϵ=

2

a

-

2

b

a

,

a≥b>0

. Choose any point

D

1

on

ℰ

. Let

D

2

be the point opposite

D

1

and let

D

be a point on

ℰ

different from

D

1

and

D

2

. Tangents to

ℰ

at

D

1

and

D

2

are parallel and meet the tangent at

D

and at points

E

1

and

E

2

, respectively. Then



D

1

E

1



D

2

E

2

=

2

a

-

2

ϵ

2

b

.

Draw a circle with center

D

1

and radius



D

1

E

1



and a circle with center

D

2

and radius



D

2

E

2



; suppose these meet in points

T

and

U

. But then



D

1

T

D

2

T=

2

a

-

2

ϵ

2

b

. Thus

T

is a point on a Cassini oval with foci

D

1

and

D

2

. The same is true for the point

U

. It can be shown that the foci

F

1

and

F

2

are also on the oval.