Unit quaternion integral curve: A new type of fair free-form curves Original Research Article

This paper proposes a new type of free-form curves for fairness. A unit quaternion curve is used to specify the tangent of the curve in order to more directly manipulate its curvature and variation of curvature than is possible for the traditional parametric representations like Bézier and NURBS curves. Since the new curve is represented by an integral form of a unit quaternion curve, it is named a unit quaternion integral: QI curve. It is a generalization and an extension of the clothoid into the three dimensional space and the norm of its tangent is always equal to 1. Its curvature and variation of curvature are given by rather simple expressions.