Ray tracing generalized cylinders
ACM Transactions on Graphics (TOG)
Two moving coordinate frames for sweeping along a 3D trajectory
Computer Aided Geometric Design
Computing frames along a trajectory
Computer Aided Geometric Design
Calculation of reference frames along a space curve
Graphics gems
Constrained optimal framings of curves and surfaces using quaternion Gauss maps
Proceedings of the conference on Visualization '98
SIGGRAPH '85 Proceedings of the 12th annual conference on Computer graphics and interactive techniques
A Predictor-Corrector Technique for Visualizing Unsteady Flow
IEEE Transactions on Visualization and Computer Graphics
Quaternion Frame Approach to Streamline Visualization
IEEE Transactions on Visualization and Computer Graphics
Faking Dynamics of Ropes and Springs
IEEE Computer Graphics and Applications
Arc-Length-Based Axial Deformation and Length Preserved Animation
CA '97 Proceedings of the Computer Animation
Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves
Computer Aided Geometric Design
Almost rotation-minimizing rational parametrization of canal surfaces
Computer Aided Geometric Design
Bender: a virtual ribbon for deforming 3D shapes in biomedical and styling applications
Proceedings of the 2005 ACM symposium on Solid and physical modeling
Geometric skinning with approximate dual quaternion blending
ACM Transactions on Graphics (TOG)
Quintic space curves with rational rotation-minimizing frames
Computer Aided Geometric Design
Swept Volume Parameterization for Isogeometric Analysis
Proceedings of the 13th IMA International Conference on Mathematics of Surfaces XIII
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
Analysis, reconstruction and manipulation using arterial snakes
ACM SIGGRAPH Asia 2010 papers
Approximate rational solutions torational ODEs defined on discrete differentiable curves
Proceedings of the 36th international symposium on Symbolic and algebraic computation
Computer Aided Geometric Design
A complete classification of quintic space curves with rational rotation-minimizing frames
Journal of Symbolic Computation
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
Transfinite surface interpolation with interior control
Graphical Models
Optimal tool orientation control for 5-axis CNC milling with ball-end cutters
Computer Aided Geometric Design
An approach for designing a developable surface through a given line of curvature
Computer-Aided Design
Skeleton-based intrinsic symmetry detection on point clouds
Graphical Models
A general and efficient method for finding cycles in 3D curve networks
ACM Transactions on Graphics (TOG)
Vector-projection approach to curve framing for extruded surfaces
ICCSA'13 Proceedings of the 13th international conference on Computational Science and Its Applications - Volume 1
Inverse kinematics for optimal tool orientation control in 5-axis CNC machining
Computer Aided Geometric Design
Rotation-minimizing osculating frames
Computer Aided Geometric Design
Vessel visualization using curvicircular feature aggregation
EuroVis '13 Proceedings of the 15th Eurographics Conference on Visualization
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Due to its minimal twist, the rotation minimizing frame (RMF) is widely used in computer graphics, including sweep or blending surface modeling, motion design and control in computer animation and robotics, streamline visualization, and tool path planning in CAD/CAM. We present a novel simple and efficient method for accurate and stable computation of RMF of a curve in 3D. This method, called the double reflection method, uses two reflections to compute each frame from its preceding one to yield a sequence of frames to approximate an exact RMF. The double reflection method has the fourth order global approximation error, thus it is much more accurate than the two currently prevailing methods with the second order approximation error—the projection method by Klok and the rotation method by Bloomenthal, while all these methods have nearly the same per-frame computational cost. Furthermore, the double reflection method is much simpler and faster than using the standard fourth order Runge-Kutta method to integrate the defining ODE of the RMF, though they have the same accuracy. We also investigate further properties and extensions of the double reflection method, and discuss the variational principles in design moving frames with boundary conditions, based on RMF.