Structural invariance of spatial Pythagorean hodographs

✍ Scribed by Rida T. Farouki; Mohammad al-Kandari; Takis Sakkalis

• Publisher: Elsevier Science
• Year: 2002
• Tongue: English
• Weight: 103 KB
• Volume: 19
• Category: Article
• ISSN: 0167-8396
• DOI: 10.1016/s0167-8396(02)00123-1

✦ Synopsis

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 factoredquaternion representation for Pythagorean hodographs in three-dimensional Euclidean space-a particular instance of the "PH representation map" proposed by Choi et al. ( )-and the unit quaternion description of spatial rotations. This approach furnishes a remarkably simple derivation for the polynomials ũ(t), ṽ(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 realtime PH curve CNC interpolators.