IBM Journal of Research and Development
An algebraic approach to curves and surfaces on the sphere and on other quadrics
Selected papers of the international symposium on Free-form curves and free-form surfaces
The conformal map z→z2 of the hodograph plane
Computer Aided Geometric Design
Hermite interpolation by Pythagorean hodograph quintics
Mathematics of Computation
Real-time CNC interpolators for Pythagorean-hodograph curves
Computer Aided Geometric Design
The elastic bending energy of Pythagorean-hodograph curves
Computer Aided Geometric Design
Minkowski pythagorean hodographs
Computer Aided Geometric Design
Construction and shape analysis of PH Hermite interpolants
Computer Aided Geometric Design
Performance analysis of CNC interpolators for time-dependent feedrates along PH curves
Computer Aided Geometric Design
Hermite interpolation by pythagorean hodograph curves of degree seven
Mathematics of Computation
Structural invariance of spatial Pythagorean hodographs
Computer Aided Geometric Design
Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves
Computer Aided Geometric Design
Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures
Computer Aided Geometric Design
Weierstrass-type approximation theorems with Pythagorean hodograph curves
Computer Aided Geometric Design
Journal of Symbolic Computation
Geometric Hermite interpolation by monotone helical quintics
Computer Aided Geometric Design
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A novel formulation for spatial Pythagorean hodograph (PH) curves, based on the geometric product of vectors from Clifford algebra, is proposed. Compared to the established quaternion representation, in which a hodograph is generated by a continuous sequence of scalings/rotations of a fixed unit vector n@?, the new representation corresponds to a sequence of scalings/reflections of n@?. The two representations are shown to be equivalent for cubic and quintic PH curves, when freedom in choosing n@? is retained for the vector formulation. The latter also subsumes the original (sufficient) characterization of spatial Pythagorean hodographs, proposed by Farouki and Sakkalis, as a particular choice for n@?. In the context of the spatial PH quintic Hermite interpolation problem, variation of the unit vector n@? offers a geometrically more-intuitive means to explore the two-parameter space of solutions than the two free angular variables that arise in the quaternion formulation. This space is seen to have a decomposition into a product of two one-parameter spaces, in which one parameter determines the arc length and the other can be used to vary the curve shape at fixed arc length.