2. Constructing a Point on a Cassini Oval

a

1

b

1

θ

0.4

zoom

3

show ellipse

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

Let

F

1

and

F

2

be two fixed points (the foci) a distance

2a

apart. A Cassini oval (or Cassini ellipse) is a quartic curve traced by a point

T

such that the product of the distances



F

1

T×

F

1

T

is a constant

2

b

.

Let

c=

2

a

+

2

b

and let

σ

1

be the circle with center

F

1

and radius

2c

. Let

M

be a point on

σ

1

and let

D

2

be the midpoint of

M

F

2

. Let

D

1

be the orthogonal projection of

F

1

on the perpendicular bisector of

M

F

2

. Let

τ

1

be the circle with center

F

1

and radius

d

1

=

F

1

D

1



; let

τ

2

be the circle with center

F

2

and radius

d

2

=

F

2

D

2



. Let

T

be the intersection of the circles

τ

1

and

τ

2

. Let

α

be the angle between

D

1

D

2

and

F

1

F

2

.

Then

2

(

d

1

+

d

2

)

+

2

(2a)

2

sin

α=

2

(2c)

,

2

(

d

1

-

d

2

)

=

2

(2a)

2

cos

α

. The difference of these two equations gives

d

1

d

2

=

2

c

-

2

a

=

2

b

, so that

T

satisfies the defining condition for the oval.