The Clothoid

Introduction:

The clothoid is interesting because its curvature varies linearly as the parameter t. It is one of the simplest examples of a curve that can be constructed from its curvature, although the math is somewhat complicated. There are also clothoids whose curvature varies as the nth power of the parameter t. An example with n = 2 is plotted below. Note the extremely simple formula for arclength.

Definition:

,

where

The definition for a clothoid of degree n is too complicated to include here.

Properties:

Velocity and Acceleration:

The Tangent Vector:

Arc Length:

Curvature: