# Constructing Quadratic Curves

Constructing Quadratic Curves

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 is on an ellipse if the sum of the distances from to two other points (the foci) and is constant. A point is on a hyperbola if the difference of the distances from to two other points (the foci) and is constant; taking the difference one way gives one branch of the hyperbola and the other way gives the other branch.

P

P

F

1

F

2

P

P

F

1

F

2

Parabola: let be the focus of the parabola, let be a point on the directrix, and let be the intersection of the perpendicular to the directrix at and the bisector of the segment , so that .

C

A

P

A

AC

AP=CP

Ellipse: let and be the foci of an ellipse, let the point be on the circle with center and radius . Let the point be the intersection of the bisector of the segment and the straight line , so that .

F

1

F

2

A

F

2

r

P

AF

1

AF

2

PF+PF=r

1

2

Hyperbola: let and be the foci of a hyperbola, let the point be on the circle with center and radius . Let the point be the intersection of the bisector of the segment and the straight line through and , so that .

F

1

F

2

A

F

2

r

P

AF

1

A

F

2

PF-PF=r

1

2

Line is always tangent to the curve at .

BP

P