# 2. Constructing a Point on a Cassini Oval

2. Constructing a Point on a Cassini Oval

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

Let and be two fixed points (the foci) a distance apart. A Cassini oval (or Cassini ellipse) is a quartic curve traced by a point such that the product of the distances is a constant .

F

1

F

2

2a

T

T×T

F

1

F

1

2

b

Let and let be the circle with center and radius . Let be a point on and let be the midpoint of . Let be the orthogonal projection of on the perpendicular bisector of . Let be the circle with center and radius =; let be the circle with center and radius =. Let be the intersection of the circles and . Let be the angle between and .

c=+

2

a

2

b

σ

1

F

1

2c

M

σ

1

D

2

M

F

2

D

1

F

1

M

F

2

τ

1

F

1

d

1

F

1

D

1

τ

2

F

2

d

2

F

2

D

2

T

τ

1

τ

2

α

D

1

D

2

F

1

F

2

Then +α=, =α. The difference of these two equations gives =-=, so that satisfies the defining condition for the oval.

2

(+)

d

1

d

2

2

(2a)

2

sin

2

(2c)

2

(-)

d

1

d

2

2

(2a)

2

cos

d

1

d

2

2

c

2

a

2

b

T