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

figure

ellipse	hyperbola
parabola	oval of Cassini

move point P

semimajor axis, a

branch of hyperbola

left

right

move focus of parabola

Cassini oval parameter, b

loop

left

right

Given two points,

F

1

and

F

2

(the foci), an ellipse is the locus of points

P

such that the sum of the distances from

P

to

F

1

and to

F

2

is a constant. A hyperbola is the locus of points

P

such that the absolute value of the difference between the distances from

P

to

F

1

and to

F

2

is a constant. An oval of Cassini is the locus of points

P

such that the product of the distances from

P

to

F

1

and to

F

2

is a constant (

2

b

here). A parabola is the locus of points

P

such that the distance from

P

to a point

F

(the focus) is equal to the distance from

P

to a line

L

(the directrix). This Demonstration illustrates those definitions by letting you move a point along the figure and watch the relevant distances change.