“Ray tracing trimmed rational surface patches” by Nishita, Sederberg and Kakimoto

  • ©Tomoyuki Nishita, Thomas (Tom) W. Sederberg, and Masanori Kakimoto




    Ray tracing trimmed rational surface patches

Session/Category Title: Surfaces and Volumes




    This paper presents a new algorithm for computing the points at which a ray intersects a rational Bézier surface patch, and also an algorithm for determining if an intersection point lies within a region trimmed by piecewise Bézier curves. Both algorithms are based on a recent innovation known as Bézier clipping, described herein. The intersection algorithm is faster than previous methods for which published performance data allow reliable comparison. It robustly finds all intersections without requiring special preprocessing.


    1. Btihm, Wolfgang, Farin, Gerald and Kahmann, Jurgen. A survey of curve and surface methods in CAGD. Computer Aided Geometric Design, 1, 1 (1984), 1-60.
    2. Farin, Gerald. Curves and Surfaces for Computer Aided Geometric Design, Academic Press, 1988.
    3. Glassner, Andrew ed., Introduction to Ray Tracing. Academic Press, 1989.
    4. Joy, Keneth and Bhetanabhotla, Murthy. Ray Tracing Parametric Patches Using Numerical Techniques and Ray Coherence. Proceedings of SIGGRAPH ’86 (Dallas, TX, August 18-22, 1986). In Computer Graphics, 20, 4 (August 1986), 279-285.
    5. Joy, Kenneth, Grant, Charles ,Max, Nelson and Hatfield, Lansing. Computer Graphics: Image Synthesis. Computer Society Press, 1988.
    6. Kajiya, Jim. Ray tracing parametric patches. Proceedings of SIGGRAPH ’82 (Boston, MA, July 26-30, 1982). In Computer Graphics,16,3 (July 1982), 245-254.
    7. Lane, Jeff and Riesenfeld, Rich. A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces. IEEE Trans. RAMI,2 (1980), 35-46.
    8. Murakami, Koichi, Hirota, Katsuhiko and Ishii, Mitsuo. Fast ray tracing. Fujitsu Science and Technology Journal, 24, 2 (1988), 150-159.
    9. Rockwood, Alyn, Heaton, Kurt and Davis, Tom. Real-Time Rendering of Trimmed Surfaces. Proceedings of SIGGRAPH ’89 (Boston, MA, July 31 – August 4, 1989). In Computer Graphics, 23, 3 (July 1989), 107-117.
    10. Rogers, Dave. Procedural Elements for Computer Graphics. McGraw-Hill, New York, 1985, 296-305.
    11. Roth, Scott. Ray Casting for Modeling Solids. Computer Graphics and Image Processing, 18, 1982, 109-144.
    12. Sederberg, Tom. An Algorithm for Algebraic Curve Intersection. Computer-Aided Design, 21, 9 (1989), 547-554.
    13. Sederberg, Tom and Parry, Scott. A Comparison of Three Curve Intersection Algorithms. Computer-Aided Design, 18, 1 (1986), 58-63.
    14. Sederberg, Tom and Parry, Scott. Free-Form Deformation of Solid Geometric Models. Proceedings of SIGGRAPH ’86 (Dallas, TX, August 18-22, 1986). In Computer Graphics, 20, 4 (August 1986), 151-160.
    15. Sederberg, Tom, White, Scott and Zundel, Alan. Fat Arcs: A Bounding Region with Cubic Convergence. Computer Aided Geometric Design, 6 (1989), 205-218.
    16. Shantz, Mike and Chang, Sheue-Ling. Rendering Trimmed NURBS with adaptive Forward Differencing. Proceedings of SIGGRAPH ’88 (Atlanta, GA, August 1-5, 1988). In Computer Graphics, 22, 4 (August 1988), 189-198.
    17. Sweeney, Michael and Bartels, Richard. Ray Tracing Free- Form B-Spline Surfaces. IEEE CG&A, 6, 2, 1986, 41-49.
    18. Toth, Dan. On Ray Tracing Parametric Surfaces. Proceedings of SIGGRAPH ’85 (San Francisco, CA, July 22-26, 1985). In Computer Graphics 19, 3 (July 1985), 171-179.
    19. Whitted, Turner. An Improved illumination Model for Shaded Display. CACM, 23, 6, 1980, 96-102.
    20. Woodward, Charles. Ray Tracing Parametric Surfaces by Subdivision in Viewing Plane. in W. Strasser and H.-P. Seidel, editors, Theory and Practice of Geometric Modeling, Springer-Verlag, 1989, 273-290.

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