# 3. Constructing a Point on a Cassini Oval

3. Constructing a Point on a Cassini Oval

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

An ellipse is given with the equation +=1 and eccentricity , . Choose any point on . Let be the point opposite and let be a point on different from and . Tangents to at and are parallel and meet the tangent at and at points and , respectively. Then .

ℰ

x

2

a

2

y

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b

2

ϵ=

a-b

2

2

a

a≥b>0

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ℰ

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2

D

1

D

ℰ

D

1

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ℰ

D

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D

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D

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DEDE=a-ϵb

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2

Draw a circle with center and radius and a circle with center and radius ; suppose these meet in points and . But then . Thus is a point on a Cassini oval with foci and . The same is true for the point . It can be shown that the foci and are also on the oval.

D

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DE

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1

D

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DE

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T

U

DTDT=a-ϵb

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T

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U

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2