B-Spline Pythagorean Hodograph Curves in Clifford Algebras

In several recent publications B-spline functions appeared with control points from abstract algebras, e.g. complex numbers, quaternions or Clifford algebras. In the context of constructions of Pythagorean hodograph curves, computations with these B-splines occur, mixing the components of the control points. In this paper we detect certain unifying patterns common to all these computations. We show that two essential components can be separated. The first one is the usual B-spline function squaring and integration, producing a new knot sequence and a new array of real coefficients for the control point computation. The second one is a special commutative multiplication which can be defined even in non-commutative algebras. We use this general Clifford algebra based approach to reconstruct some known results for the signatures (2, 0), (3, 0) and (2, 1) and add a new construction for the signature (3, 1). This last case is essential for the description of canal surfaces. It is shown that Clifford algebra is an especially suitable tool for the general description of B-spline curves with Pythagorean hodograph property. The presented unifying definition of PH B-splines is general and is not limited to any particular knot sequences or control points. In a certain sense, this paper can be considered as a continuation of the 2002 article by Choi et al. with regard to the B-splines.