document: Structural invariance of spatial Pythagorean hodographs The structural invariance of the four-polynomial characterization for three-dimensional Pythagorean hodographs introduced by Dietz et al. -LRB- 1993 -RRB- , 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. -LRB- 2002 -RRB- - and the unit quaternion description of spatial rotations . This approach furnishes a remarkably simple derivation for the polynomials u -LRB- t -RRB- , upsilon -LRB- t -RRB- , p -LRB- t -RRB- , q -LRB- t -RRB- that specify the canonical form of a rotated Pythagorean hodograph , in terms of the original polynomials u -LRB- t -RRB- , upsilon -LRB- t -RRB- , p -LRB- t -RRB- , q -LRB- t -RRB- and the angle theta 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
doc_bio_tags: B I O B I I O B I O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O B I I O B I O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O O B I I O O O O O O O O O O O O O O O B I O O O O O O O O O O O O O