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
Curves with rational Frenet-Serret motion
Computer Aided Geometric Design
Structural invariance of spatial Pythagorean hodographs
Computer Aided Geometric Design
Characterization and construction of helical polynomial space curves
Journal of Computational and Applied Mathematics
Computer Aided Geometric Design
A characterization of quintic helices
Journal of Computational and Applied Mathematics
Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures
Computer Aided Geometric Design
Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves
Journal of Symbolic Computation
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Pythagorean-Hodograph Curves: Algebra and Geometry Inseparable
Spatial pythagorean hodograph quintics and the approximation of pipe surfaces
IMA'05 Proceedings of the 11th IMA international conference on Mathematics of Surfaces
Salkowski curves revisited: A family of curves with constant curvature and non-constant torsion
Computer Aided Geometric Design
Helical polynomial curves and double Pythagorean hodographs II. Enumeration of low-degree curves
Journal of Symbolic Computation
Quintic space curves with rational rotation-minimizing frames
Computer Aided Geometric Design
Rational rotation-minimizing frames on polynomial space curves of arbitrary degree
Journal of Symbolic Computation
Advances in Computational Mathematics
Geometric Hermite interpolation by monotone helical quintics
Computer Aided Geometric Design
Rational Pythagorean-hodograph space curves
Computer Aided Geometric Design
Cubic helical splines with Frenet-frame continuity
Computer Aided Geometric Design
Geometric design using space curves with rational rotation-minimizing frames
MMCS'08 Proceedings of the 7th international conference on Mathematical Methods for Curves and Surfaces
Pythagorean-hodograph curves in Euclidean spaces of dimension greater than 3
Journal of Computational and Applied Mathematics
Rotation-minimizing osculating frames
Computer Aided Geometric Design
Hi-index | 0.01 |
For regular polynomial curves r(t) in R^3, relations between the helicity condition, existence of rational Frenet frames, and a certain ''double'' Pythagorean-hodograph (PH) structure are elucidated in terms of the quaternion and Hopf map representations of spatial PH curves. After reviewing the definitions and properties of these representations, and conversions between them, linear and planar PH curves are identified as degenerate spatial PH curves by certain linear dependencies among the coefficients. Linear and planar curves are trivially helical, and all proper helical polynomial curves are PH curves. All spatial PH cubics are helical, but not all PH quintics. The two possible types of helical PH quintic (monotone and general) are identified as subsets of the PH quintics by constraints on their quaternion coefficients. The existence of a rational Frenet frame and curvature on polynomial space curves is equivalent to a certain ''double'' PH form, first identified by Beltran and Monterde, in which |r^'(t)| and |r^'(t)xr^'^'(t)| are both polynomials in t. All helical PH curves are double PH curves, which encompass all PH cubics and all helical PH quintics, although non-helical double PH curves of higher order exist. The ''double'' PH condition is thoroughly analyzed in terms of the quaternion and Hopf map forms, and their connections. A companion paper presents a complete characterization of all helical and non-helical double PH curves up to degree 7.