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Ray Tracing Free-Form B-Spline Surfaces
February 1986 (vol. 6 no. 2)
pp. 41-49
Michael Sweeney, Abel Image Research
Richard Bartels, University of Waterloo
We present a method for using ray tracing to render spline surfaces?one that is suitable for any object generated from control vertices via tensor-product B-splines. The method derives from kajiya's work on ray tracing procedurally defined surfaces1 and makes use of two preprocessing steps. One involves the controlvertex refinement recurrences due to Riesenfeld et. al.2, and the second generates a tree of nested bounding boxes. Intersection testing involves running Kajiya's algorithm on the tree, followed by two to three (on the average) iterations of Newton's method.

1. James T.Kajiya, "New Techniques for Rav Tracing Procedurally Defined Objects," Trans. Graphics. Vol. 2, No. 3, pp. l61-181 July 1983
2. RichardRiesenfeld, ElaineCohen, and TomLyche, "Discrete B-Splines and Subdivision Techniques in Computer-Aided Geometric Design and Computer Graphics," Computer Graphics and Image Processing Vol. 14, No. 2, pp. 87-111 Oct. 1980
3. JohnAmanatides, "Ray Tracing with Cones," Computer Graphics (Proc. SIGGRAPH 84) July 1984 Vol. 18, No. 3, pp. 129-136
4. Alan H.Barr, "Global and Local Deformations of Solid Primitives," Computer Graphics (Proc. SIGGRAPH 83) July 1983 Vol. 17, No. 3, (State-of-the-Art in Image Synthesis Seminar Notes)
5. Brian ABarsky, Tony D.DeRose, and L. RichardSpeer, "A Theoretical and Empirical Analysis of Coherent Ray-Tracing," Proc. Graphics Interface 85 May 1985pp. 1-8
6. S.Coquillart and M.Gangnet, "Shaded Display of Digital Maps," IEEE Trans. Computer Graphics and Applications Vol. 3, No. 7, pp. 35-42 July 1984
7. Robert L.Cook, ThomasPorter, and LorenCarpenter, "Distributed Ray Tracing," Computer Graphics (Proc. SIGGRAPH 84) July 1984 Vol. 18, No. 3, pp. 137-146
8. MarkDippe and JohnSwensen, "An Adaptive Subdivision Algorithm and Parallel Architecture for Realistic Image Synthesis," Computer Graphics (Proc. SIGGRAPH 84) July 1984 Vol. 18, No. 3, pp. 149-158
9. A.S.Glassner, "Space Subdivision for Fast Ray Tracing," IEEE Trans. Computer Graphics and Applications Vol. 4, No. 10, pp. 15-22 Oct. 1984
10. Roy A.Hall and Donald P.Greenberg, "A Testbed for Realistic Image Synthesis," IEEE Computer Graphics and Applications Vol. 3, No. 8, pp. 10-20 Nov. 1983
11. Roy A.Hall, "A Methodology or Realistic Image Synthesis," 1983
12. PatrickHanrahan, "Ray Tracing Algebraic Surfaces," Computer Graphics(Proc. SIGGRAPH 83) July 1983 Vol. 17, No. 3, pp. 83-90
13. Paul S.Heckbert and PatrickHanrahan, "Beam Tracing Polygonal Objects," Computer Graphics (Proc. SIGGRAPH 84) July 1984 Vol. 18, No. 3, pp. 119-128
14. James T.Kajiya, "Ray Tracing Parametric Patches," Computer Graphics (Proc. SIGGRAPH 82) July 1982 Vol. 16, No. 3, pp. 245-254
15. James T.Kajiya and Brian P. VoniHerzen, "Ray Tracing Volume Densities," Computer Graphics (Proc. SIGGRAPH 84) July 1984 Vol. 18, No. 3, pp. 165-174
16. Douglas S.Kay, "Transparency, Refraction and Ray Tracing for Computer Synthesized Images," January 1979
17. MichaelPotmesil and IndranilChakravarty, "Synthetic Image Generation with a Lens and Aperture Camera Model," ACM Trans. Graphics Vol. 1, No. 2, pp. 85-108 April 1982
18. Scott D.Roth, "Ray Casting as a Method for Solid Modeling," , General Motors Research Laboratories 1980
19. Scott D.Roth, "Ray Casting as a Method for Solid Modeling, Computer Vision," Computer Graphics and Image Processing Vol. 18, No. 2, pp. 109-144 Feb. 1980
20. Steven M.Rubin and Turner A.Whitted, "Three-Dimensional Representation for Fast Rendering of Complex Scenes," Computer Graphics (Proc. SIGGRAPH 80) July 1980 Vol. 14, No. 3, pp. 110-116
21. Thomas W.Sederberg and David C.Anderson, "Ray Tracing Steiner Patches," Computer Graphics (Proc. SIGGRAPH 84) July 1984 Vol. 18, No. 3, pp. 159-164
22. Daniel L.Toth, "On Ray Tracing Parametric Surfaces," Computer Graphics(Proc. SIGGRAPH 85) July 1985 Vol. 19, No. 3, pp. 171-179
23. HankWeghorst, GaryHooper, and Donald P.Greenberg, "Improved Computational Methods for Ray Tracing," ACM Trans. Graphics Vol. 3, No. 1, pp. 52-69 Jan. 1984
24. Turner A.Whitted, "An Improved Model for Shaded Display," Comm. ACM Vol. 23, No. 6, pp. 343-349 June 1980
25. Michael A.J.Sweeney, "The Waterloo CGL Ray-Tracing Package," 1984
26. Richard F.Riesenfeld, "Applications of B-Spline Approximation to Geometric Problems of Computer-Aided Design," 1973
27. CarldeBoor, A Practical Guide to Splines , Springer-Verlag 1978pp. 392 Applied Mathematical Sciences 27
28. Brian A.Barsky, "End Conditions and Boundary Conditions for Uniform B-spline Curve and Surface Representations," Computers in Industry Vol. 3, pp. 17-29 Mar. 1982 The Steven A. Coons Memorial Issue
29. Brian A.Barsky and John C.Beatty, "Controlling the Shape of Parametric B-Spline and Beta-Spline Curves," Proc. Graphics Interface 83 May 1983pp. 223-232
30. Richard H.Bartels, John C.Beatty, and Brian A.Barsky, "An Introduction to the Use of Splines in Computer Graphics," , Dept. of Computer Science, University of Waterloo pp. 324 (revised Apr. 1985) Dept. of Electrical Engineering, University of California
31. Bui TuongPhong, "Illumination for Computer Generated Pictures," Comm. ACM Vol. 18, No. 6, pp. 311-317 June 1975
32. Robert L.Cook and Kenneth E.Torrance, "A Reflection Model for Computer Graphics," Computer Graphics (Proc. SIGGRAPH 81) Aug. 1981 Vol. 15, No. 3, pp. 307-316

Michael Sweeney, Richard Bartels, "Ray Tracing Free-Form B-Spline Surfaces," IEEE Computer Graphics and Applications, vol. 6, no. 2, pp. 41-49, Feb. 1986, doi:10.1109/MCG.1986.276691
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