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WOLFRAM|DEMONSTRATIONS PROJECT

1. Constructing a Point on a Cassini Oval

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a
0.76
b
0.81
α
0.31
zoom
1.2
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show α
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.
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