# Ellipse and Friends

Ellipse and Friends

The table shows the equations in rectangular coordinates of an ellipse in standard position, as well as its two associated hyperbolas and their asymptotes.

2 x 2 a 2 y 2 b | ellipse |

- 2 x 2 a 2 y 2 b | left–righthyperbola |

2 x 2 a 2 y 2 b | up–downhypberbola |

y b x a | asymptotes |

The asymptotes are a limiting case of both kinds of hyperbolas. For example, if , then the up-down hyperbola has equation -=.000000001≈0, so that (, and one or the other (or both) , , whose graphs are two straight lines.

a=b=.00001

2

x

2

y

x+y)(x-y)≈0

x+y≈0

x-y≈0

The lemniscate is included for two reasons:

The lemniscate has a property similar to the ellipse. Let Q and R be two points, the foci. For a point P on the ellipse, |PQ| + |PR| = , a constant. For a point P on the lemniscate, |PQ| × |PR| = , a constant.

c

k

Also, the arc lengths (partial or full perimeters) of the ellipse and the lemniscate are related. The arc length of the ellipse is calculated using an incomplete elliptic integral of the second kind, while the arc length of the lemniscate is given by an elliptic integral of the first kind.