“Ray tracing parametric patches”

  • ©James (Jim) T. Kajiya




    Ray tracing parametric patches



    This paper describes an algorithm that uses ray tracing techniques to display bivariate polynomial surface patches. A new intersection algorithm is developed which uses ideas from algebraic geometry to obtain a numerical procedure for finding the intersection of a ray and a patch without subdivision. The algorithm may use complex coordinates for the (u, v)-parameters of the patches. The choice of these coordinates makes the computations more uniform, so that there are fewer special cases to be considered. In particular, the appearance and disappearance of silhouette edges can be handled quite naturally. The uniformity of these techniques may be suitable for implementation on either a general purpose pipelined machine, or on special purpose hardware.


    1. APPEL, A., “Some Techniques for Shading Machine Renderings of Solids”, 1968 SJCC, pp. 37-45.
    2. BLINN, J.F., “Computer Display of Curved Surfaces” Ph.D. Thesis, U. of Utah, Computer Science Department, Salt Lake City, Utah. 1978.
    3. BLINN, J.F., “Simulation of Wrinkled Surfaces”, Computer Graphics, v.12, August 1978, pp.286-292.
    4. BLINN, J.F., CARPENTER, L.C., LANE, J.M. AND WHITTED, T., “Scan Line Methods for Displaying Parametrically Defined Surfaces”, Comm. ACM, v.23, January 1980, pp.23-34.
    5. CATMULL, E.E., “A Subdivision Algorithm for Computer Display of Curved Surfaces” Ph.D. Thesis, U. of Utah, Computer Science Department, Salt Lake City, Utah. 1974.
    6. LANE, J.M., AND CARPENTER, L.C., “A Generalized Scan Line Algorithm for the Computer Display of Parametrically Defined Surfaces”, Computer Graphics and Image Processing, v.11, 1979, pp.290-297.
    7. LITTLEWOOD, D.E., A University Algebra, Dover, 1970.
    8. RALSTON, A., A First Course in Numerical Analysis, McGraw-Hill 1965.
    9. ROTH, S.D., “Ray Casting for Modeling Solids” Computer Graphics and Image Processing, v.18, 1982, pp.109-144.
    10. RUBIN, S., AND WHITTED, T., “A Three-Dimensional Representation for Fast Rendering of Complex Scenes”, Computer Graphics, v.14, 1980, pp.110-116.
    11. ULLNER, M., private communication, 1981.
    12. USPENSKY, J.V., Theory of Equations, McGraw-Hill, 1948.
    13. WALKER, R.J., Algebraic Curves, Springer-Verlag, 1950.
    14. WHITTED, T., “An Improved Illumination Model for Shaded Display”, Comm. ACM, v.23, June 1980, pp.343-349.

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