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
Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves
Computer Aided Geometric Design
Computation of rotation minimizing frames
ACM Transactions on Graphics (TOG)
Nonexistence of rational rotation-minimizing frames on cubic curves
Computer Aided Geometric Design
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
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
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
Original Articles: Motion design with Euler-Rodrigues frames of quintic Pythagorean-hodograph curves
Mathematics and Computers in Simulation
Pythagorean-hodograph curves in Euclidean spaces of dimension greater than 3
Journal of Computational and Applied Mathematics
An interpolation scheme for designing rational rotation-minimizing camera motions
Advances in Computational Mathematics
Rotation-minimizing osculating frames
Computer Aided Geometric Design
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A rotation-minimizing adapted frame on a space curve r(t) is an orthonormal basis (f"1,f"2,f"3) for R^3 such that f"1 is coincident with the curve tangent t=r^'/|r^'| at each point and the normal-plane vectors f"2, f"3 exhibit no instantaneous rotation about f"1. Such frames are of interest in applications such as spatial path planning, computer animation, robotics, and swept surface constructions. Polynomial curves with rational rotation-minimizing frames (RRMF curves) are necessarily Pythagorean-hodograph (PH) curves-since only the PH curves possess rational unit tangents-and they may be characterized by the fact that a rational expression in the four polynomials u(t), v(t), p(t), q(t) that define the quaternion or Hopf map form of spatial PH curves can be written in terms of just two polynomials a(t), b(t). As a generalization of prior characterizations for RRMF cubics and quintics, a sufficient and necessary condition for a spatial PH curve of arbitrary degree to be an RRMF curve is derived herein for the generic case satisfying u^2(t)+v^2(t)+p^2(t)+q^2(t)=a^2(t)+b^2(t). This RRMF condition amounts to a divisibility property for certain polynomials defined in terms of u(t), v(t), p(t), q(t) and their derivatives.