This Article 
 Bibliographic References 
 Add to: 
Cutting and Stitching: Converting Sets of Polygons to Manifold Surfaces
April-June 2001 (vol. 7 no. 2)
pp. 136-151

Abstract—Many real-world polygonal surfaces contain topological singularities that represent a challenge for processes such as simplification, compression, and smoothing. We present an algorithm that removes singularities from nonmanifold sets of polygons to create manifold (optionally oriented) polygonal surfaces. We identify singular vertices and edges, multiply singular vertices, and cut through singular edges. In an optional stitching operation, we maintain the surface as a manifold while joining boundary edges. We present two different edge stitching strategies, called pinching and snapping. Our algorithm manipulates the surface topology and ignores physical coordinates. Except for the optional stitching, the algorithm has a linear complexity and requires no floating point operations. In addition to introducing new algorithms, we expose the complexity (and pitfalls) associated with stitching. Finally, several real-world examples are studied.

[1] S. Doo and M. Sabin, “Analysis of the Behaviour of Recursive Division Surfaces Near Extraordinary Points,” Computer Aided Design, vol. 10, no. 6, pp. 356-360, 1978.
[2] A. Varshney, "Hierarchical Geometric Approximations," PhD thesis TR-050-1994, Dept. of Computer Science, Univ. of North Carolina, Chapel Hill, 1994.
[3] A. Guéziec, Locally Toleranced Surface Simplification IEEE Trans. Visualization and Computer Graphics, vol. 5, no. 2, pp. 168-189, Apr.-June 1999.
[4] G. Taubin et al., “Geometry Coding and VRML,” Proc. IEEE, vol. 86, no. 6, pp. 1228-1243, June 1998.
[5] H. Hoppe, “Efficient Implementation of Progressive Meshes,” Technical Report MSR-TR-98-02, Microsoft Research, Redmond, Wash., Jan. 1998.
[6] J.H. Bohn, “Removing Zero-Volume Parts from CAD Models for Layered Manufacturing,” IEEE Computer Graphics and Applications, vol. 15, no. 6, pp. 27-34, Nov. 1995.
[7] G. Taubin, "A Signal Processing Approach to Fair Surface Design," Computer Graphics Proc., Ann. Conf. Series, ACM Siggraph, ACM Press, New York, 1995, pp.351-358.
[8] F. Harary, Graph Theory. Addison-Wesley, 1969.
[9] M.K. Agoston, Algebraic Topology: A First Course. New York: Marcel Dekker, 1976.
[10] X. Sheng and I.R. Meier, "Generating Topological Structures for Surface Models," IEEE Computer Graphics and Applications, vol. 15, no. 6, pp. 35-41, Nov. 1995.
[11] G. Barequet and S. Kumar, "Repairing CAD Models," Proc. IEEE Visualization, pp. 363-370,Phoenix, Ariz., 1997.
[12] A. Hubeli and M. Gross, “Fairing of Non-Manifolds for Visualization,” Proc. 11th Ann. IEEE Visualization Conf. (Vis) 2000, 2000.
[13] A.P. Gueziec, G. Taubin, F. Lazarus, and W. Horn, “Converting Sets of Polygons to Manifold Surfaces by Cutting and Stitching,” Proc. IEEE Visualization '98, pp. 383-390, Oct. 1998.
[14] J. Rossignac and D. Cardoze, "Matchmaker: Manifold Breps for Non-Manifold r-sets," Technical Report: GIT-GVU-99-03, GVU Center, Georgia Inst. of Tech nology, Oct. 1998, . To appear in the Proc. ACM Symp. Solid Modeling, 1999.
[15] T.K. Dey, H. Edelsbrunner, S. Guha, and D. Nekhayev, “Topology Preserving Edge Contraction,” Publications de l'Institut Mathematique (Beograd), vol. 60, no. 80, 1999, also, Technical Report RGI-Tech-98-018, Raindrop Geomagic Inc. , Research Triangle Park, N.C., 1998.
[16] H. Hoppe, T. DeRose, T. Duchamp, J. McDonald, and W. Stuetzle, “Mesh Optimization,” Proc. SIGGRAPH '93, pp. 19-26, 1993.
[17] A. Guéziec, “Surface Simplification with Variable Tolerance,” Proc. Second Ann. Int'l Symp. Medical Robotics and Computer Assisted Surgery, pp. 132-139, Nov. 1995.
[18] M. Segal and C.H. Sequin, “Partitioning Polyhedral Objects into Non-Intersecting Parts,” IEEE Computer Graphics and Applications, vol. 8, no. 1, pp. 53-67, Jan. 1988.
[19] SLA CAD, Stereolithography Interface Specification, 3D Systems Inc., Valencia, Calif., 1989.
[20] S.J. Rock and M.J. Wozny, “Generating Topological Information from a 'Bucket of Facets,'” Solid Freeform Fabrication Symp. Proc., H.L. Marcus, ed., pp. 251-259, Aug. 1992.
[21] I. Makela and A. Dolenc, “Some Efficient Procedures for Correcting Triangulated Models,” Proc. Symp. Solid Freeform Fabrication, pp. 126-134, July 1993.
[22] M.C. Bailey, “Tele-Manufacturing: Rapid Prototyping on the Internet,” IEEE Computer Graphics and Applications, vol. 15, no. 6, pp. 20-26, Nov. 1995.
[23] V. Chandru, S. Manohar, and C.E. Prakash, “Voxel-Based Modeling for Layered Manufacturing,” IEEE Computer Graphics and Applications, vol. 15, no. 6, pp. 42-47, Nov. 1995.
[24] J.H. Bøhn and M.J. Wozny, "A Topology-Based Approach for Shell-Closure," Geometric Modeling for Product Realization, P.R. Wilson et al., eds., pp. 297-319. North-Holland, 1993.
[25] J.H. Bohn, “Automatic CAD-Model Repair,” PhD thesis, Rensselaer Polytechnic Inst., Troy, N.Y., Aug. 1993.
[26] G. Barequet and M. Sharir, "Filling Gaps in the Boundary of a Polyhedron," Computer Aided Geometric Design, vol. 12, no. 2, pp. 207-229, 1995.
[27] G. Barequet, “Using Geometric Hashing to Repair CAD Objects,” IEEE Computational Science and Eng., vol. 4, no. 4, pp. 22-28, Oct. 1997.
[28] G. Butlin and C. Stops, “CAD Data Repair,” Proc. Fifth Int'l Meshing Roundtable, pp. 7-12, 1996.
[29] T. Murali and T. Funkhouser, "Consistent Solid and Boundary Representations from Arbitrary Polygonal Data," Proc. Symp. Interactive 3D Graphics, pp. 155-162,Providence, R.I., 1997.
[30] R. Szeliski, D. Tonnesen, and D. Terzopoulos, “Curvature and Continuity Control in Particle-Based Surface Models,” Geometric Methods in Computer Vision II, vol. 2031-15, pp. 172-181, July 1993.
[31] W. Welch and A. Witkin, “Free-Form Shape Design Using Triangulated Surfaces,” Siggraph '94 Conf. Proc., pp. 247-256, July 1994.
[32] P. Veron and J.C. Leon, “Static Polyhedron Simplification Using Error Measurements,” Computer Aided Design, vol. 29, no. 4, pp. 287-298, Apr. 1997.
[33] C.M. Hoffmann, Geometric and Solid Modeling, Morgan Kaufmann, San Mateo, Calif., 1989.
[34] A.A.G. Requicha,“Representations for rigid solids: Theory, methods, and systems,” ACM Computing Surveys, vol. 12, no. 4, pp. 437-464, 1980.
[35] H. Desaulniers and N.F. Stewart, “An Extension of Manifold Boundary Representations to the r-Sets,” ACM Trans. Graphics, vol. 11, no. 1, pp. 40-60, Jan. 1992.
[36] C.M. Hoffmann, J.E. Hopcroft, and M.S. Karasick, “Robust Set Operations on Polyhedral Solids,” IEEE CG&A, Vol. 9, No. 6, Nov. 1989, pp. 50-59.
[37] J.A. Heisserman, “Generative Geometric Design and Boundary Solid Grammars,” PhD thesis, Carnegie Mellon Univ., May 1991.
[38] G. Abram and L. Treinish, "An Extended Data-Flow Architecture for Data Analysis and Visualization," IEEE Visualization 95 Conf. Proc., IEEE Computer Society Press, Los Alamitos, Calif., Oct. 1995, pp. 263-270.
[39] Open Visualization Data Explorer, /, 1999.
[40] B.G. Baumgart, Modeling for Computer Vision, PhD thesis, Dept. of Computer Science, Stanford University, Palo Alto, Calif., 1974.
[41] A. Kalvin, “Segmentation and Surface-Based Modeling of Objects in Three-Dimensional Biomedical Images,” PhD thesis, New York Univ., June 1991.
[42] The Virtual Reality Modeling Language Specification, Version 2.0, June 1997,
[43] T.H. Cormen,C.E. Leiserson, and R.L. Rivest,Introduction to Algorithms.Cambridge, Mass.: MIT Press/McGraw-Hill, 1990.

Index Terms:
Polygonal surface, topological singularities, manifold, cutting, stitching.
André Guéziec, Gabriel Taubin, Francis Lazarus, Bill Horn, "Cutting and Stitching: Converting Sets of Polygons to Manifold Surfaces," IEEE Transactions on Visualization and Computer Graphics, vol. 7, no. 2, pp. 136-151, April-June 2001, doi:10.1109/2945.928166
Usage of this product signifies your acceptance of the Terms of Use.