Constructing Quadratic Curves

parabola

ellipse

hyperbola

Drag the point A.

directrix

0.5

A

C

B

P

This Demonstration constructs the parabola, ellipse, and hyperbola geometrically. These constructions only need a straightedge and compass.

Here are the geometric definitions of these curves. A parabola is the set of points equidistant from a line (the directrix) and a point (the focus). A point

P

is on an ellipse if the sum of the distances from

P

to two other points (the foci)

F

1

and

F

2

is constant. A point

P

is on a hyperbola if the difference of the distances from

P

to two other points (the foci)

F

1

and

F

2

is constant; taking the difference one way gives one branch of the hyperbola and the other way gives the other branch.

Parabola: let

C

be the focus of the parabola, let

A

be a point on the directrix, and let

P

be the intersection of the perpendicular to the directrix at

A

and the bisector of the segment

AC

, so that

AP=CP

.

Ellipse: let

F

1

and

F

2

be the foci of an ellipse, let the point

A

be on the circle with center

F

2

and radius

r

. Let the point

P

be the intersection of the bisector of the segment

AF

1

and the straight line

AF

2

, so that

PF

1

+PF

2

=r

.

Hyperbola: let

F

1

and

F

2

be the foci of a hyperbola, let the point

A

be on the circle with center

F

2

and radius

r

. Let the point

P

be the intersection of the bisector of the segment

AF

1

and the straight line through

A

and

F

2

, so that

PF

1

-PF

2

=r

.

Line

BP

is always tangent to the curve at

P

.