“Hierarchical B-spline refinement” by Forsey and Bartels

Refinement is usually advocated as a means of gaining finer control over a spline curve or surface during editing. For curves, refinement is a local process. It permits the change of control vertices and subsequent editing of the detail in one region of the curve while leaving control vertices in other regions unaffected. For tensor-product surfaces, however, refinement is not local in the sense that it causes control vertices far from a region of interest to change as well as changing the control vertices that influence the region. However, with some care and understanding it is possible to restrict the influence of refinement to the locality at which an editing effect is desired. We present a method of localizing the effect of refinement through the use of overlays, which are hierarchically controlled subdivisions. We also introduce two editing techniques that are effective when using overlays: one is direct surface manipulation through the use of edit points and the other is offset referencing of control vertices.

1. Barr, Alan, “Global and Local Deformations of Solid Primitives”, Proceedings of SIGGRAPH’84 (Minneapolis, Minnesota, July 23-27, 1984). In Computer Graphics. 18,3 (July, 1984) 21-30.
2. Bartels, Richard, Beatty, John, and Barsky, Brian, An Introduction to Splines for Use in Computer Graphics and Geometric Modeling, Morgan Kaufmann Publishers, Palo Alto, California (1987).
3. Cobb, Elizabeth, Design of Sculptured Sulfaces Using the B-Spline Representation, University of Utah PhD Thesis, Salt Lake City, Utah (1984).
4. Cohen, Elaine, Lyche, Tom, and Riesenfeld, Richard, “Discrete B-spfines and Subdivision Techniques in Computer-Aided Geometric Design and Computer Graphics,” Computer Graphics and Image Processing, 14,2 (October, 1980) 87-111.
5. Faux, Ivor and Pratt, Michael, Computational Geometry for Design and Manufacture, John Wiley & Sons (1979).
6. Foley, James and van Dam, Andries, Fundamentals of Interactive Computer Graphics, Addison Wesley (1982).
7. Forsey, David and Wilhelrns, Jane, “Techniques for Interactive Manipulation of Articulated Bodies Using Dynamic Analysis,” Proceedings of Graphics Interface ’88 to appear} (1988).
8. Joe, Barry, “Discrete Beta-Splines,” Proceedings of SIGGRAPH’87 (Anaheim, California, July 27-31, 1987). In Computer Graphics, 21,4 (July, 1987) 137-144.
9. Mortenson, Michael, Geometric Modeling, Wiley, New York (1985).
10. Sederberg, Tom and Parry, Scott, “Free-Form Deformation of Solid Geometric Models,” Proceedings of SIGGRAPH’86 (Dallas, Texas, August 18-22, 1986). In Computer Graphics, 20,4 (August, 1986) 151-160.