The Spiral of Archimedes

Introduction:

The spiral plotted above is one branch of a curve known as the spiral of Archimedes. In fact, it is part of a family of spirals, many of which have different names. Some examples are given below.

Definition:

,

where a is the radius of the spiral, and n is some real number. In the case n = 1, plotted at the top of the page, the spiral is called the spiral of Archimedes. The family of spirals also has the polar parameterization

Several values of n other than 1 are also given specific names:

Spiral of Archimedes (n = 1) Fermat's Spiral (n = 2)
Hyperbolic Spiral (n = -1) Lituus (n = -2)

 

Properties:

Velocity and Acceleration:

The Tangent Vector:

Arc Length:

Curvature: