A new type of free-form curve given by an integral form

The paper proposes a new type of free-form curve 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 Bezier and NURBS curves. Since the new curve is represented by can integral form of a unit quaternion curve, it is named unit quaternion integral curve or QI curve for brevity. It is a generalization and an extension of the clothoid into 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.