Wenping Wang
Yang Liu
Abstract
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.
- Banks, D. C. and Singer, B. A. 1995. A predictor-corrector technique for visualizing unsteady flows. IEEE Trans. on Visualiz. Comput. Graph. 1, 2, 151--163. Google Scholar
Digital Library
- Barzel, R. 1997. Faking dynamics of ropes and springs. IEEE Comput. Graph. Appl. 17, 3, 31--39. Google ScholarDigital Library
- Bechmann, D. and Gerber, D. 2003. Arbitrary shaped deformation with dogme. Visual Comput. 19, 2--3, 175--186.Google ScholarCross Ref
- Bishop, R. L. 1975. There is more than one way to frame a curve. Amer. Math. Monthly 82, 3, 246--251.Google ScholarCross Ref
- Bloomenthal, J. 1985. Modeling the mighty maple. In Proceedings of SIGGRAPH. 305--311. Google ScholarDigital Library
- Bloomenthal, J. 1990. Calculation of reference frames along a space curve. In A. Glassner Ed. Graphics Gems, Academic Press. New York, NY. Google ScholarDigital Library
- Bloomenthal, M. and Riesenfeld, R. F. 1991. Approximation of sweep surfaces by tensor product NURBS. In SPIE Proceedings Curves and Surfaces in Computer Vision and Graphics II. Vol. 1610. 132--154.Google Scholar
- Bronsvoort, W. F. and Flok, F. 1985. Ray tracing generalized cylinders. ACM Trans. Graph. 4, 4, 291--302. Google ScholarDigital Library
- Choi, H. I., Kwon, S.-H., and Wee, N.-S. 2004. Almost rotation-minimizing rational parametrization of canal surfaces. Comput. Aid. Geomet. Des. 21, 9, 859--881. Google ScholarDigital Library
- Chung, T. L. and Wang, W. 1996. Discrete moving frames for sweep surface modeling. In Proceedings of Pacific Graphics. 159--173.Google Scholar
- Farouki, R. 2002. Exact rotation-minimizing frames for spatial Pythagorean-hodograph curves. Graphic. Models 64, 382--395. Google ScholarDigital Library
- Farouki, R. and Han, C. Y. 2003. Rational approximation schemes for rotation-minimizing frames on Pythagorean-hodograph curves. Comput. Aid. Geomet. Des. 20, 7, 435--454. Google ScholarDigital Library
- Goemans, O. and Overmars, M. 2004. Automatic generation of camera motion to track a moving guide. In Proceedings of Workshop on the Algorithmic Foundations of Robotics. (WAFR) 201--216.Google Scholar
- Guggenheimer, H. W. 1989. Computing frames along a trajectory. Comput. Aid. Geomet. Des. 6, 77--78. Google ScholarDigital Library
- Hanson, A. 1998. Constrained optimal framing of curves and surfaces using quaternion gauss map. In Proceedings of Visulization. 375--382. Google ScholarDigital Library
- Hanson, A. 2005. Visualizing Quaternions. Morgan Kaufmann.Google Scholar
- Hanson, A. J. and Ma, H. 1995. A quaternion approach to streamline visualization. IEEE Trans. Visualiz. Comput. Graph. 1, 2, 164--174. Google ScholarDigital Library
- Jüttler, B. 1998. Rotational minimizing spherical motions. In Advacnes in Robotics: Analysis and Control. Kluwer, 413--422.Google Scholar
- Jüttler, B. 1999. Rational approximation of rotation minimizing frames using Pythagorean-hodograph cubics. J. Geom. Graph. 3, 141--159.Google Scholar
- Jüttler, B. and Mäurer, C. 1999. Cubic Pythagorean hodograph spline curves and applications to sweep surface modeling. Comput.-Aid. Des. 31, 73--83.Google ScholarCross Ref
- Klok, F. 1986. Two moving frames for sweeping along a 3D trajectory. Comput. Aid. Geomet. Des. 3, 1, 217--229. Google ScholarDigital Library
- Kreyszig, E. 1991. Differential Geometry. Dover.Google Scholar
- Lazarus, F., Coquillart, S., and Jancène, P. 1993. Interactive axial deformations. In Modeling in Computer Graphics. Springer Verlag, 241--254.Google Scholar
- Lazarus, F. and Verroust, A. 1994. Feature-based shape transformation for polyhedral objects. In Proceedings of the 5th Eurographics Workshop on Animation and Simulation. 1--14.Google Scholar
- Lazarus, S. C. and Jancene, P. 1994. Axial deformation: an intuitive technique. Comput.-Aid. Des. 26, 8, 607--613.Google ScholarCross Ref
- Llamas, I., Powell, A., Rossignac, J., and Shaw, C. 2005. Bender: a virtual ribbon for deforming 3d shapes in biomedical and styling applications. In Proceedings of Symposium on Solid and Physical Modeling. 89--99. Google ScholarDigital Library
- Peng, Q., Jin, X., and Feng, J. 1997. Arc-length-based axial deformation and length preserving deformation. In Proceedings of Computer Animation. 86--92. Google ScholarDigital Library
- Poston, T., Fang, S., and Lawton, W. 1995. Computing and approximating sweeping surfaces based on rotation minimizing frames. In Proceedings of the 4th International Conference on CAD/CG. Wuhan, China.Google Scholar
- Pottmann, H. and Wagner, M. 1998. Contributions to motion based surface design. Int. J. Shape Model. 4, 3&4, 183--196.Google ScholarCross Ref
- Semwal, S. K. and Hallauer, J. 1994. Biomedical modeling: implementing line-of-action algorithm for human muscles and bones using generalized cylinders. Comput. Graph. 18, 1, 105--112.Google ScholarCross Ref
- Shani, U. and Ballard, D. H. 1984. Splines as embeddings for generalized cylinders. Comput. Vision Graph. Image Proces. 27, 129--156.Google ScholarCross Ref
- Siltanen, P. and Woodward, C. 1992. Normal orientation methods for 3D offset curves, sweep surfaces, skinning. In Proceedings of Eurographics. 449--457.Google Scholar
- Wang, W. and Joe, B. 1997. Robust computation of rotation minimizing frame for sweep surface modeling. Comput.-Aid Des. 29, 379--391.Google ScholarCross Ref
- Wang, W., Jüttler, B., Zheng, D., and Liu, Y. 2007. Computation of rotation minimizing frame in computer graphics. Tech. rep., TR 2007-07, Department of Computer Science, University of Hong Kong.Google Scholar
Index Terms
- Computation of rotation minimizing frames
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