Computation of Rotation Minimizing Frame in Computer Graphics

We investigate the computation and properties of rotation minimizing frame (RMF), which is a moving orthonormal frame U(u) attached to a smooth curve x(u), called the spine curve, in 3D such that U(u) does not rotate about the instantaneous tangent of x(u). Due to its minimal-twist property, the 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. In general, the RMF cannot be computed exactly and therefore one often needs to approximate the exact RMF by a sequence of orthonormal frames at sampled points on the spine curve. We present a novel simple and efficient method for accurate and stable computation of an RMF for any C 1 regular 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 is highly accurate – it has the global fourth order approximation error, thus comparing favorably to the second order approximation error of two currently prevailing methods – the projection method by Klok and the rotation method by Bloomenthal, while all these methods have comparable per-frame computational cost. Furthermore, the double reflection method is much simpler and faster than using the standard 4-th order Runge-Kutta method to integrate the defining ODE of the RMF, which yields the same accuracy as the double reflection method. We also present further properties and extensions of the double reflection method for various application scenarios. Finally, we discuss the variational principles in design moving frames with boundary conditions, based on the RMF.

[1]  J. Bloomenthal.  Calculation of reference frames along a space curve , 1990 .

[2]  Bert Jüttler,et al.  Cubic Pythagorean hodograph spline curves and applications to sweep surface modeling , 1999, Comput. Aided Des..

[3]  Shiaofen Fang,et al.  Computing and approximating sweepingsurfaces based on rotation minimizing framesTim , 1995 .

[4]  Rida T. Farouki,et al.  Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves , 2003, Comput. Aided Geom. Des..

[5]  Jieqing Feng,et al.  Arc-Length-Based Axial Deformation and Length Preserving Deformation , 1998 .

[6]  Barry Joe,et al.  Robust computation of the rotation minimizing frame for sweep surface modeling , 1997, Comput. Aided Des..

[7]  Jules Bloomenthal,et al.  Modeling the mighty maple , 1985, SIGGRAPH.

[8]  Hyeong In Choi,et al.  Almost rotation-minimizing rational parametrization of canal surfaces , 2004, Comput. Aided Geom. Des..

[9]  Francis Lazarus,et al.  Feature-based shape transformation for polyhedral objects , 1994 .

[10]  Dominique Bechmann,et al.  Arbitrary shaped deformations with DOGME , 2003, Vis. Comput..

[11]  Fopke Klok.  Two moving coordinate frames for sweeping along a 3D trajectory , 1986, Comput. Aided Geom. Des..

[12]  Helmut Pottmann,et al.  Contributions to Motion Based Surface Design , 1998, Int. J. Shape Model..

[13]  Rida T. Farouki,et al.  Exact rotation-minimizing frames for spatial Pythagorean-hodograph curves , 2002, Graph. Model..

[14]  Andrew J. Hanson,et al.  Visualizing quaternions , 2005, SIGGRAPH Courses.

[15]  Michael E. Taylor,et al.  Differential Geometry I , 1994 .

[16]  Ronen Barzel,et al.  Faking Dynamics of Ropes and Springs , 1997, IEEE Computer Graphics and Applications.

[17]  H. GUGGENHEIMER.  Computing frames along a trajectory , 1989, Comput. Aided Geom. Des..

[18]  Bert Jüttler.  Rotation Minimizing Spherical Motions , 1998 .

[19]  Constrained optimal framings of curves and surfaces using quaternion Gauss maps , 1998, Proceedings Visualization '98 (Cat. No.98CB36276).

[20]  Sabine Coquillart,et al.  Interactive Axial Deformations , 1993, Modeling in Computer Graphics.

[21]  Richard F. Riesenfeld,et al.  Approximation of sweep surfaces by tensor product NURBS , 1992, Other Conferences.

[22]  Andrew J. Hanson,et al.  Quaternion Frame Approach to Streamline Visualization , 1995, IEEE Trans. Vis. Comput. Graph..

[23]  David C. Banks,et al.  A Predictor-Corrector Technique for Visualizing Unsteady Flow , 1995, IEEE Trans. Vis. Comput. Graph..

[24]  Pekka Siltanen,et al.  Normal orientation methods for 3D offset curves, sweep surfaces and skinning , 1992, Comput. Graph. Forum.

[25]  Willem F. Bronsvoort,et al.  Ray tracing generalized cylinders , 1985, TOGS.

[26]  Dana H. Ballard,et al.  Splines as embeddings for generalized cylinders , 1982 .

[27]  R. Bishop.  There is More than One Way to Frame a Curve , 1975 .

[28]  B. Jüttler,et al.  Rational approximation of rotation minimizing frames using Pythagorean-hodograph cubics , 1999 .

[29]  Sudhanshu Kumar Semwal,et al.  Biomechanical modeling: Implementing line-of-action algorithm for human muscles and bones using generalized cylinders , 1994, Comput. Graph..

[30]  Sabine Coquillart,et al.  Axial deformations: an intuitive deformation technique , 1994, Comput. Aided Des..