This Article 
 Bibliographic References 
 Add to: 
Ray-Tracing Triangular Trimmed Free-Form Surfaces
July-September 1998 (vol. 4 no. 3)
pp. 202-214

Abstract—This paper presents a new approach to rendering triangular algebraic free-form surfaces. A hierarchical subdivision of the surface with associated tight bounding volumes provides for quick identification of the surface regions likely to be hit by a ray. For each leaf of the hierarchy, an approximation to the corresponding surface region is stored. The approximation is used to compute a good starting point for the iteration, which ensures rapid convergence. Trimming curves are described by a tree of trimming primitives, such as squares, circles, polygons, and free-form curves, combined with Boolean operations. For trimmed surfaces, an irregular adaptive subdivision is constructed to quickly eliminate all parts outside the trimming curve from consideration during rendering. Cost heuristics are introduced to optimize the rendering time further.

[1] D.H. Ballard, "Strip Trees: A Hierarchical Representation for Curves," Comm. ACM, vol. 24, no. 5, pp. 310-321, 1981.
[2] W. Barth and W. Stürzlinger, "Efficient Ray-tracing for Bézier and B-Spline Surfaces," Computers&Graphics, vol. 17, no. 4, pp. 423-430, July 1993.
[3] J. Bramsteidl, "Ray-tracing von Getrimmten Freiformflächen," Master's thesis, Univ. of Linz, Austria, Mar. 1996.
[4] S. Campagna, "Vergleich und Erweiterung von Verfahren für die Schnittpunktberechnung von Strahlen mit Polynomflächen," Master's thesis, Univ. of Erlangen, Germany, Aug. 1995.
[5] W. Dahmen, C.A. Micchelli, and H.-P. Seidel, "Blossoming Begets B-Splines Built Better by B-Patches," Math. Comp., vol. 59, pp. 97-115, 1992.
[6] G.E. Farin, “Triangular Bernstein-Bézier Patches,” Computer Aided Geometric Design, pp. 83-127, Mar. 1986.
[7] G.E. Farin, Curves and Surfaces for Computer Aided Geometric Design: A Practical Guide, fourth ed. Academic Press, 1996.
[8] D.J. Filip and R.N. Goldman, "Conversion from Bézier-Rectangles to Bézier-Triangles," Computer Aided Design, vol. 19, pp. 25-27, 1987.
[9] J.D. Foley,A. van Dam,S.K. Feiner,, and J.F. Hughes,Computer Graphics: Principles and Practice,Menlo Park, Calif.: Addison-Wesley, 1990.
[10] A. Fournier and J. Buchanan, "Chebyshev Polynomials for Boxing and Intersections of Parametric Curves and Surfaces," Computer Graphics Forum (Proc. EUROGRAPHICS '94), vol. 13, no. 3, pp. 127-142, Sept. 1994.
[11] R.N. Goldman, "Subdivision Algorithms for Bézier Triangles," Computer Aided Design, vol. 15, pp. 159-166, 1983.
[12] S. Gottschalk, M. Lin, and D. Manocha, "Obb-Tree: A Hierarchical Structure for Rapid Interference Detection," Proc. ACM Siggraph '96, pp. 171-180, 1996.
[13] B. Hamann and P.-Y. Tsai, "A Tessellation Algorithm for the Representation of Trimmed NURBS Surfaces with Arbitrary Trimming Curves," Computer-Aided Design, vol. 28, pp. 461-472, 1996.
[14] J.T. Kajiya, "New Techniques for Ray-tracing Procedurally Defined Objects," Computer Graphics (Proc. SIGGRAPH '83), vol. 17, no. 3, pp. 91-102, July 1983.
[15] S. Krishnan and D. Manocha, "An Efficient Surface Intersection Algorithm based on Lower Dimensional Formulation," ACM Trans. Computer Graphics, vol. 16, no. 1, pp. 74-106, 1997.
[16] T. Nishita, T.W. Sederberg, and M. Kakimoto, "Ray-tracing Trimmed Rational Surface Patches," Computer Graphics (Proc. SIGGRAPH '90), vol. 24, pp. 337-345, Aug. 1990.
[17] K. Qin, M. Gong, Y. Guan, and W. Wang, "A New Method for Speeding Up Ray-tracing NURBS Surfaces," Computers&Graphics, vol. 21, no. 5, pp. 577-586, 1997.
[18] S.M. Rubin and T. Whitted, "A 3-Dimensional Representation for Fast Rendering of Complex Scenes," Computer Graphics, vol. 14, pp. 110-116, July 1980.
[19] W. Stürzlinger, R.F. Tobler, and M. Schindler, "FLIRT—Faster Than Light Ray Tracer," Technical Report, Institut für Computergraphik, Technical Univ. of Vienna, Aug. 1993.
[20] M. Sweeney and R. Bartels, "Ray-tracing Free-Form B-Spline Surfaces," IEEE Computer Graphics and Applications, vol. 6, no. 2, pp. 41-49, 1986.
[21] D. Toth, "On Ray-tracing Parametric Surfaces," Computer Graphics (Proc. SIGGRAPH '85), vol. 19, pp. 171-179, July 1985.
[22] J. Yen, S. Spach, M. Smith, and R. Pulleyblank, "Parallel Boxing in B-Spline Intersection," IEEE Computer Graphics and Applications, vol. 11, no. 1, pp. 72-79, Jan. 1991.

Index Terms:
Ray-tracing, free-form surface, NURBS, trimming curve, tesselation, triangulation, CAD.
Wolfgang Stürzlinger, "Ray-Tracing Triangular Trimmed Free-Form Surfaces," IEEE Transactions on Visualization and Computer Graphics, vol. 4, no. 3, pp. 202-214, July-Sept. 1998, doi:10.1109/2945.722295
Usage of this product signifies your acceptance of the Terms of Use.