Structural invariance of spatial Pythagorean hodographs

Abstract

The structural invariance of the four-polynomial characterization for three-dimensional Pythagorean hodographs introduced by Dietz et al. (1993), under arbitrary spatial rotations, is demonstrated. The proof relies on a factored-quaternion representation for Pythagorean hodographs in three-dimensional Euclidean space—a particular instance of the “PH representation map” proposed by Choi et al. (2002)—and the unit quaternion description of spatial rotations. This approach furnishes a remarkably simple derivation for the polynomials u ̃ (t) , v ̃ (t) , p ̃ (t) , q ̃ (t) that specify the canonical form of a rotated Pythagorean hodograph, in terms of the original polynomials u( t), v( t), p( t), q( t) and the angle θ and axis n of the spatial rotation. The preservation of the canonical form of PH space curves under arbitrary spatial rotations is essential to their incorporation into computer-aided design and manufacturing applications, such as the contour machining of free-form surfaces using a ball-end mill and real-time PH curve CNC interpolators.