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.

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

.

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

.

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

.

Line

is always tangent to the curve at

.