This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Accurate Solid Modeling Using Polyhedral Approximations
May/June 1988 (vol. 8 no. 3)
pp. 14-28

Although curved-surface solid modeling systems achieve a higher level of accuracy than faceted systems, they also introduce a host of topological, geometric, and numerical complications. A method for calculating accurate boundary representations of solid models is introduced that reduces the impact of these complications. The method uses a pair of bounding polyhedral approximations to enclose the boundary of each object. A structural analysis automatically determines where to make adaptive refinements to the polyhedrons to assure the topological validity of the results. Potential singularities are localized. The implementation is an experimental extension to the Geometric Design Processor (GDP) solid modeling system.

1. A.A.G. Requicha, "Representation of Rigid Solid Objects,"Computer Aided Design, J. Encarnacao, ed., Springer-Verlag, New York, 1980, pp. 2-78.
2. A. A. Requicha, "Representations for rigid solids: Theory, methods, and systems,"Comput. Surveys, vol. 12, no. 4, pp. 437-465, 1980.
3. M. Mantyla and R. Sulonen, "GWB: A Solid Modeler with Euler Operations,"CG&A, Sept. 1982, pp. 17-31.
4. B.G. Baumgart, "Winged Edge Polyhedron Representation," Stanford Artificial Intelligence Lab., Memo AIM-179 and Stanford Univ. Computer Science Report No. STAN-CS-320, Oct. 1972.
5. I. Braid, "The Synthesis of Solids Bounded by Many Faces,"CACM, Apr. 1975, pp. 209-216.
6. A.A.G. Requicha and H.B. Voelcker, "Boolean Operations in Solid Modeling: Boundary Evaluation and Merging Algorithms,"Proc. IEEE, Jan. 1985, pp. 30-44.
7. H.U. Pfeifer, "Methods Used for Intersecting Geometrical Entities in the GPM Module for Volume Geometry,"Computer-Aided Design, Sept. 1985, pp. 311-318.
8. G. Allen, "Testing the Accuracy of Solid Models,"Computer-Aided Engineering, June 1985, pp. 50-54.
9. M.A. Wesley et al., "A Geometric Modeling System for Automated Mechanical Assembly,"IBM J. Research and Development, Jan. 1980, pp. 64-74.
10. W. Fitzgerald, F. Gracer, and R. Wolfe, "GRIN: Interactive Graphics for Modeling Solids,"IBM J. Research and Development, July 1981, pp. 281-294.
11. J.G. Griffiths, "A Data-Structure for the Elimination of Hidden Surfaces by Patch Subdivision,"Computer-Aided Design, July 1975, pp. 171-178.
12. J.G. Griffiths, "A Surface Display Algorithm,"Computer-Aided Design, Jan. 1978, pp. 65-73.
13. S.L. Hanna, J.F. Abel, and D.P. Greenberg, "Intersection of Parametric Surfaces by Means of Look-Up Tables,"CG&A, Oct. 1983, pp. 39-48.
14. J.R. Rossignac and A.A.G. Requicha, "Offsetting Operations in Solid Modelling," Tech. Memo. No. 53, Production Automation Project, Univ. of Rochester, 1985.
15. J.M. Lane and R.F. Riesenfeld, "A Theoretical Development for the Computer Generation and Display of Piecewise Polynomial Surfaces,"IEEE Trans. Pattern Analysis and Machine Intelligence, Jan. 1980, pp. 35-46.
16. W.E. Carlson, "An Algorithm and Data Structure for 3D Object Synthesis Using Surface Patch Intersections,"Computer Graphics(Proc. SIGGRAPH), July 1982, pp. 255-263.
17. E.G. Houghton et al., "Implementation of a Divide-and-Conquer Method for Intersection of Parametric Surfaces,"Computer-Aided Geometric Design, Vol. 2, 1985, pp. 173-183.
18. Computer-Graphics Aided Three-Dimensional Interactive Application (CATIA), user manual, SH20-2629, IBM Corp., Poughkeepsie, N.Y., 1985.
19. C.M. Brown, "PADL-2: A Technical Summary,"CC&A, Mar. 1982, pp. 69-84.
20. A.L. Klosterman, "A Geometric Modeler Based on a Dual-Geometry Representation--Polyhedrons and Rational B-Splines,"NASA Symp. Computer-Aided Geometry Modeling, NASA Langley Research Center, Hampton, Va., 1983, pp. 7-9.
21. P.R. Wilson, "Solid Modeling Research and Applications in the USA,"Geometric Modeling for CAD Applications, M. Wozny, J. Encarnacao, and H. McLaughlin, eds., North-Holland, Amsterdam, 1988.
22. A.A.G. Requicha and H.B. Voelcker, "Solid Modeling: A Historical Summary and Contemporary Assessment,"CG&A, Mar. 1982, pp. 9-24.
23. A.A.G. Requicha and H.B. Voelcker, "Solid Modeling: Current Status and Research Directions,"CG&A, Oct. 1983, pp. 25-37.
24. G.A. Crocker and W.F. Reinke, "Boundary Evaluation of Non-Convex Primitives to Produce Parametric Trimmed Surfaces,"Computer Graphics(Proc. Siggraph), Vol. 21, No. 4, July 1987, pp. 129-136.
25. J.U. Turner, "Precise Solid Modeling with a Faceted Modeler," IBM Tech. Report No. TR 00.3365, Poughkeepsie, N.Y., 1985.
26. C.M. Eastman and K. Weiler, "Geometric Modeling Using the Euler Operators,"First Ann. Conf. Computer Graphics in CAD/CAM Systems, MIT, Cambridge, Mass., May 1979.
27. P.R. Wilson, "Euler Formulas and Geometric Modeling,"CG&A, Aug. 1985, pp. 24-36.
28. I. Faux and M. Pratt,Computational Geometry for Design and Manufacture. Ellis Horwood, 1979.
29. W. Boehm, G. Farin, and J. Kahmann, "A survey of curve and surface methods in CAGD,"Comput. Aided Geometric Des., vol. 1, no. 1, pp. 1-60, July 1984.
30. T.W. Sederberg and R.N. Goldman, "Algebraic Geometry for Computer-Aided Geometric Design,"CG&A, June 1986, pp. 52-59.
31. T.W. Sederberg and S.R. Parry, "Comparison of Three Curve Intersection Algorithms,"Computer-Aided Design, Jan./Feb. 1986, pp. 58-63.
32. J.U. Turner, "Topological Matching of Boundary Models," IBM Tech. Report No. TR 00.3481, Poughkeepsie, N.Y., 1987.
33. J. Stoer and R. Bulirsch,Introduction to Numerical Analysis, Springer-Verlag, New York, 1980.
34. P.J. Deuflhard, "A Modified Newton Method for the Solution of Ill-Conditioned Systems of Nonlinear Equations with Application to Multiple Shooting,"Numerical Mathematics, 1974, pp. 289-315.
35. J.V. Uspensky,Theory of Equations, McGraw-Hill, New York, 1948.

Citation:
Joshua U. Turner, "Accurate Solid Modeling Using Polyhedral Approximations," IEEE Computer Graphics and Applications, vol. 8, no. 3, pp. 14-28, May-June 1988, doi:10.1109/38.510
Usage of this product signifies your acceptance of the Terms of Use.