Two moving coordinate frames for sweeping along a 3D trajectory
Computer Aided Geometric Design
Computing frames along a trajectory
Computer Aided Geometric Design
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
Curves with rational Frenet-Serret motion
Computer Aided Geometric Design
Modern computer algebra
Performance analysis of CNC interpolators for time-dependent feedrates along PH curves
Computer Aided Geometric Design
Structural invariance of spatial Pythagorean hodographs
Computer Aided Geometric Design
Algorithms for partial fraction decomposition and rational function integration
SYMSAC '71 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation
Algebraic factoring and rational function integration
SYMSAC '76 Proceedings of the third ACM symposium on Symbolic and algebraic computation
Aspects of symbolic integration and simplification of exponential and primitive functions.
Aspects of symbolic integration and simplification of exponential and primitive functions.
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
Almost rotation-minimizing rational parametrization of canal surfaces
Computer Aided Geometric Design
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
Quintic space curves with rational rotation-minimizing frames
Computer Aided Geometric Design
Almost rotation-minimizing rational parametrization of canal surfaces
Computer Aided Geometric Design
Computer Aided Geometric 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
Computer Aided Geometric Design
Construction of rational curves with rational rotation-minimizing frames via möbius transformations
MMCS'08 Proceedings of the 7th international conference on Mathematical Methods for Curves and Surfaces
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
Optimal tool orientation control for 5-axis CNC milling with ball-end cutters
Computer Aided Geometric Design
Construction of low degree rational motions
Journal of Computational and Applied Mathematics
C1 Hermite interpolation with spatial Pythagorean-hodograph cubic biarcs
Journal of Computational and Applied Mathematics
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An exact specification of the rotation-minimizing frame on a spatial Pythagorean-hodograph (PH) curve can be derived by integration of a rational function. The result is an angular function θ( t ) of the curve parameter, comprising in general both rational and logarithmic terms, that specifies the orientation of the rotation-minimizing frame relative to the Frenet frame. For PH cubics and quintics, the solution employs only arithmetic operations on the curve coefficients and some complex square and cube root extractions. Moreover, the generalization to PH curves of arbitrary order entails only standard polynomial algorithms (i.e., arithmetic, greatest common divisors, and resultants), solution of a linear system, and a minimal element of polynomial root-solving. Rotation-minimizing frames are employed in computer animation, the construction of swept surfaces, and in robotics applications where the axis of a tool or probe should remain tangential to a given spatial path while minimizing changes of orientation about this axis.