1. Constructing a Point on a Cassini Oval

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.

Details

The construction can be found in[2, pp. 189–190].

References

[1] E. W. Weisstein. "Cassini Ovals" from Wolfram MathWorld—A Wolfram Web Resource. mathworld.wolfram.com/CassiniOvals.html.

[2] A. A. Savelov, Plane Curves (in Croatian), Zagreb: Školska knjiga, 1979.

External Links

Cassini Ovals (Wolfram MathWorld)

Cassini Ovals

Cassini Ovals and Other Curves

Locus of Points Definition of an Ellipse, Hyperbola, Parabola, and Oval of Cassini

Families of Figure-Eight Curves

Permanent Citation

Izidor Hafner, Marko Razpet

"1. Constructing a Point on a Cassini Oval" from the Wolfram Demonstrations Project http://demonstrations.wolfram.com/1ConstructingAPointOnACassiniOval/
Published: March 29, 2018