The conformal map z→z2 of the hodograph plane
Computer Aided Geometric Design
Hermite interpolation by Pythagorean hodograph quintics
Mathematics of Computation
Construction and shape analysis of PH Hermite interpolants
Computer Aided Geometric Design
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
Characterization and construction of helical polynomial space curves
Journal of Computational and Applied Mathematics
Computer Aided Geometric Design
Topological criterion for selection of quintic Pythagorean-hodograph Hermite interpolants
Computer Aided Geometric Design
Spatial pythagorean hodograph quintics and the approximation of pipe surfaces
IMA'05 Proceedings of the 11th IMA international conference on Mathematics of Surfaces
Journal of Symbolic Computation
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
On control polygons of quartic Pythagorean--hodograph curves
Computer Aided Geometric Design
Construction of rational surface patches bounded by lines of curvature
Computer Aided Geometric Design
Advances in Computational Mathematics
Geometric Hermite interpolation by monotone helical quintics
Computer Aided Geometric Design
Rational Pythagorean-hodograph space curves
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
Design of C2 spatial pythagorean-hodograph quintic spline curves by control polygons
Proceedings of the 7th international conference on Curves and Surfaces
Original Articles: Motion design with Euler-Rodrigues frames of quintic Pythagorean-hodograph curves
Mathematics and Computers in Simulation
On the approximation order of a space data-dependent PH quintic Hermite interpolation scheme
Computer Aided Geometric Design
An interpolation scheme for designing rational rotation-minimizing camera motions
Advances in Computational Mathematics
C1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs
Journal of Computational and Applied Mathematics
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The problem of specifying the two free parameters that arise in spatial Pythagorean-hodograph (PH) quintic interpolants to given first-order Hermite data is addressed. Conditions on the data that identify when the ''ordinary'' cubic interpolant becomes a PH curve are formulated, since it is desired that the selection procedure should reproduce such curves whenever possible. Moreover, it is shown that the arc length of the interpolants depends on only one of the parameters, and that four (general) helical PH quintic interpolants always exist, corresponding to extrema of the arc length. Motivated by the desire to improve the fairness of interpolants to general data at reasonable computational cost, three selection criteria are proposed. The first criterion is based on minimizing a bivariate function that measures how ''close'' the PH quintic interpolants are to a PH cubic. For the second criterion, one of the parameters is fixed by first selecting interpolants of extremal arc length, and the other parameter is then determined by minimizing the distance measure of the first method, considered as a univariate function. The third method employs a heuristic but efficient procedure to select one parameter, suggested by the circumstances in which the ''ordinary'' cubic interpolant is a PH curve, and the other parameter is then determined as in the second method. After presenting the theory underlying these three methods, a comparison of empirical results from their implementation is described, and recommendations for their use in practical design applications are made.