“Casting Curved Shadows on Curved Surfaces” by Williams

Shadowing has historically been used to increase the intelligibility of scenes in electron microscopy and aerial survey. Various methods have been published for the determination of shadows in computer synthesized scenes. The display of shadows may make the shape and relative position of objects in such scenes more comprehensible; it is a technique lending vividness and realism to computer animation.

To date, algorithms for the determination of shadows have been restricted to scenes constructed of planar polygons. A simple algorithm is described which utilizes Z-buffer visible surface computation to display shadows cast by objects modelled of smooth surface patches. The method can be applied to all environments, in fact, for which visible surfaces can be computed. The cost of determining the shadows associated with each light source is roughly twice the cost of rendering the scene without shadows, plus a fixed transformation overhead which depends on the image resolution. No extra entities are added to the scene description in the shadowing process. This comprehensive algorithm, which permits curved shadows to be cast on curved surfaces, is contrasted with a less costly method for casting the shadows of the environment on a single ground plane.

In order to attain good results, the discrete nature of the visible-surface computations must be treated with care. The effects of dither, interpolation, and geometric quantization at different stages of the shadowing algorithm are examined. The special problems posed by self-shadowing surfaces are described.

1. Catmull, E., “A Subdivision Algorithm for Computer Display of Curved Surfaces,” PhD. thesis, Dept. of Computer Science, University of Utah, 1974. 
2. Newell, M. G., Newell, R. G., and Sancha, T. L., “A Solution to the Hidden Surface Problem,” Proceedings of the 1972 ACM National Conference. 
3. Williams, L., forthcoming PhD. thesis, University of Utah.
4. For the application of this representation to scene analysis, see: Levine, M. D., O’Handley, D. A., and Yagi, G. M., “Computer Determination of Depth Maps,” Computer Graphics and Image Processing, No. 2, 1973.
5. Crow, F. C., “Shadow Algorithms for Computer Graphics,” Siggraph 1977 Proceedings, Vol. 11, No. 2, Summer 1977. 
6. Blinn, J. F., “A Scan-Line Algorithm for the Display of Bicubic Surface Patches,” PhD. thesis, Dept. of Computer Science, university of Utah, 1978.
7. Crow, F. C., “The Aliasing Problem in Computer-Synthesized Shaded Images,” PhD. thesis, Dept. of Computer Science, University of Utah, 1976. 
8. Freeman, H., “Computer Processing of Line Drawing Images,” ACM Computing Surveys, Vol. 6, No. 1, March 1974. 
9. Blinn, J. F., “Models of Light Reflection for Computer Synthesized Pictures,” Siggraph 1977 Proceedings, Vol. 11, No. 2, Summer 1977. 
10. Sutherland, I.E., “A Head-Mounted Three-Dimensional Display,” Fall Joint Computer Conference 1968, Thompson Books, Washington, D.C., 757.