1. Constructing a Point on a Cassini Oval

a

0.76

b

0.81

α

0.31

zoom

1.2

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

Fix two points,

F

1

and

F

2

(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

σ

be the circle with center at the center of the oval and radius

a

. Let

A

2

be the right apex of the oval. A ray from

A

2

at an angle

α

to the line

F

1

F

2

meets

σ

at the points

N

1

and

N

2

. Let

τ

1

be the circle with center

F

1

and radius



A

2

N

1



and let

τ

2

be the circle with center

F

2

and radius



A

2

N

2



. Let the point

T

be one of the intersections of

τ

1

and

τ

2

. Then the product of the radii



F

1

T

and



F

2

T

is equal to the product



A

2

F

2

×

A

2

F

1

=

2

b

, so

T

is on the oval.