The Astroid

Introduction:

The astroid shown above results from cubing the coordinates of the unit circle. It is part of a more general family of parametrically and implicitly defined curves given by the equations below. Like the deltoid, it is a special case of the hypocycloid.

Definition:

Implicitly, the curve is defined by:

The curve plotted above is for the value n = 2.

Properties:

Velocity and Acceleration:

The Tangent Vector:

Arc Length:

Curvature: