Two moving coordinate frames for sweeping along a 3D trajectory
Computer Aided Geometric Design
Computing frames along a trajectory
Computer Aided Geometric Design
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
Real-time CNC interpolators for Pythagorean-hodograph curves
Computer Aided Geometric Design
The elastic bending energy of Pythagorean-hodograph curves
Computer Aided Geometric Design
Circle and sphere as rational splines
Neural, Parallel & Scientific Computations - computer aided geometric design
The circle as a smoothly joined BR-curve on [0,1]
Computer Aided Geometric Design
Rational blending surfaces between quadrics
Computer Aided Geometric Design
Curves with rational Frenet-Serret motion
Computer Aided Geometric Design
Performance analysis of CNC interpolators for time-dependent feedrates along PH curves
Computer Aided Geometric Design
A Menagerie of Rational B-Spline Circles
IEEE Computer Graphics and Applications
Structural invariance of spatial Pythagorean hodographs
Computer Aided Geometric Design
Computer Aided Geometric Design
Computation of rotation minimizing frames
ACM Transactions on Graphics (TOG)
C1 Hermite interpolation with simple planar PH curves by speed reparametrization
Computer Aided Geometric Design
Identification of spatial PH quintic Hermite interpolants with near-optimal shape measures
Computer Aided Geometric Design
Nonexistence of rational rotation-minimizing frames on cubic curves
Computer Aided Geometric Design
Quintic space curves with rational rotation-minimizing frames
Computer Aided Geometric Design
Computer aided design of ventilation tubes for customized hearing aid devices
Computer-Aided Design
Construction of rational surface patches bounded by lines of curvature
Computer Aided Geometric Design
Rational rotation-minimizing frames on polynomial space curves of arbitrary degree
Journal of Symbolic Computation
Advances in Computational Mathematics
Rational Pythagorean-hodograph space curves
Computer Aided Geometric Design
Computer Aided Geometric Design
A complete classification of quintic space curves with rational rotation-minimizing frames
Journal of Symbolic Computation
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
Spatial pythagorean hodograph quintics and the approximation of pipe surfaces
IMA'05 Proceedings of the 11th IMA international conference on Mathematics of Surfaces
Original Articles: Motion design with Euler-Rodrigues frames of quintic Pythagorean-hodograph curves
Mathematics and Computers in Simulation
Optimal tool orientation control for 5-axis CNC milling with ball-end cutters
Computer Aided Geometric Design
Rotation-minimizing osculating frames
Computer Aided Geometric Design
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An adapted frame (t, u, v) on a space curve r(ξ) is a right-handed set of three orthonormal vectors, where t is the unit tangent and u, v span the curve normal plane. For such frames to have a rational dependence on the curve parameter, r(ξ) must be a Pythagorean-hodograph (PH) curve, since only PH curves have rational unit tangent vectors. Among all possible adapted frames, the rotation-minimizing frame (RMF) is the most attractive for applications such as animation, swept surface constructions, and motion planning. The PH curves admit exact RMF descriptions, but they involve transcendental (logarithmic) functions. Since rational forms are generally preferred, the problem of rational approximation of RMFs for PH curves is considered herein. This is accomplished by employing the Euler-Rodrigues frame (ERF) as a reference (the ERF is rational and, unlike the Frenet frame, does not suffer indeterminacies at inflections). The function that describes the angular deviation between the RMF and ERF is derived in closed form, and is approximated by Padé (rational Hermite) interpolation. In typical cases, these interpolants furnish compact approximations of excellent accuracy, amenable to use in a variety of applications.