The Ellipse

Introduction:

The ellipse is a generalization of the circle. It has an interesting geometric definition (shown below).

Definition:

Implicitly, the ellipse is defined by

Geometrically, the ellipse is defined by attaching a long string to two points (called the foci) and tracing the kink in the string as it is pulled tight in a full 360 degree rotation about the foci.

Properties:

Velocity and Acceleration:

The Tangent Vector:

Arc Length:

Curvature: