Computational differential topology

Denis Blackmore, Yuriy Mileyko

Abstract

Some of the more differential aspects of the nascent field of computational topology are introduced and treated in considerable depth. Relevant categories based upon stratified geometric objects are proposed, and fundamental problems are identified and discussed in the context of both differential topology and computer science. New results on the triangulation of objects in the computational differential categories are proven, and evaluated from the perspective of effective computability (algorithmic solvability). In addition, the elements of innovative, effectively computable approaches for analyzing and obtaining computer generated representations of geometric objects based upon singularity/stratification theory and obstruction theory are formulated. New methods for characterizing complicated intersection sets are proven using differential analysis and homology theory. Also included are brief descriptions of several implementation aspects of some of the approaches described, as well as applications of the results in such areas as virtual sculpting, virtual surgery, modeling of heterogeneous biomaterials, and high speed visualizations.


Keywords

Varieties; Embeddings; Shape; Isotopy; Effectively computable; decidable; sl-projective varieties; Stratification; Homology; Obstructions

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References

K. Abdel-Malek and H. J. Yeh, On the determination of starting points for parametric surface intersections, CAD 29 (1997), 21–35.

K. Abdel-Malek and H. J. Yeh, Geometric representation of the swept volume using Jacobian rank deficiency conditions, CAD 29 (1997), 457–468.

K. Abdel-Malek, H. J. Yeh and S. Othman, Swept volumes: void and boundary identification, CAD 30 (1998), 1009–1018.

K. Abdel-Malek, W. Seaman and H. J. Yeh, NC verification of up to 5-axis machining processes using manifold stratification, ASME J. Manufacturing Sci. and Engineering 122 (2000), 121–135.

K. Abdel-Malek, J. Yang and D. Blackmore, On swept volume formulations: implicit surfaces, CAD 33 (2001), 113–121.

K. Abdel-Malek, D. Blackmore and K. Joy, Swept volumes: foundations, perspectives and applications, Int. J. Shape Modeling (submitted).

K. Abe et al., Computational topology for reconstruction of surfaces with boundary: integrating experiments and theory, in Proceedings of the IEEE International Conference on Shape Modeling and Applications, June 15 ˆu 17, 2005, Cambridge, MA, IEEE Computer Society, Los Alimitos, CA, 288 - 297.

K. Abe et al., Computational topology for isotopic surface reconstruction, Theoretical Computer Science, Special Issue – Spatial Representation: Discrete vs. Continuous Computational Models, Edited by R. Kopperman, P. Panangaden, M.B. Smyth, D. Spreen and J. Webster, 365 (3), 184–198, 2006.

P. Agarwal, H. Edelsbrunner and Y. Wang, Computing the writhing number of a polygonal knot, Discrete Comput. Geom. 32 (2004), 37–53. http://dx.doi.org/10.1007/s00454-004-2864-x

N. Amenta, S. Choi, T. Dey and N. Leekha, A simple algorithm for homeomorphic surface reconstruction, in ACM Symposium on Computational Geometry, 2000, pp. 213-222.

N. Amenta, S. Choi and R. Kolluri, The power crust, union of balls and the medial axis transform, Comput. Geom.: Theory and Applicatons 19 (2001), 127–173. http://dx.doi.org/10.1016/S0925-7721(01)00017-7

N. Amenta, T. Peters and A. Russell, Computational topology: ambient isotopic approximation of 2-manifolds (invited paper), Theoretical Computer Sci. 305 (2003), 3–15. http://dx.doi.org/10.1016/S0304-3975(02)00691-6

L.-E. Andersson, S. Dorney, T. Peters and N. Stewart, Polyhedral perturbations that preserve topological form, CAGD 12 (1995), 785–799.

L.-E. Andersson, T. Peters and N. Stewart, Equivalence of topological form for curvilinear geometric objects, Int. J. Computational Geom. and Appls. 10 (6) (2000), 609–622. http://dx.doi.org/10.1142/S0218195900000346

V. Arnold, S. Gusein-Zade and A. Varchenko, Singularities of Differentiable Maps, Vols. I & II, Birkh¨auser, Boston, 1985.

U. Axen and H. Edelsbrunner, Auditory Morse analysis of triangulated manifolds, in Mathematical Visualization, H.-C. Hege and K. Polthier (eds.), Springer-Verlag, Berlin, 1988, pp. 223-236.

C. Bajaj, C. Hoffmann, R. Lynch and J. Hopcroft, Tracing surface intersections, CAGD 5 (1988), 285–307.

S. Basu, Computing the Betti numbers of arrangements via spectral sequences, J. Computer and System Sciences 67 (2003), 244–262. http://dx.doi.org/10.1016/S0022-0000(03)00009-6

M. Bern, et al., Emerging Challenges in Computational Topology, Report of NSF Workshop on Computational Topology, June 11-12, 1999, Miami Beach, FL, http://xxx.lanl.gov/abs/cs/9909001.

J. Bisceglio, C. Mow and T. Peters, Boolean algebraic operands for engineering design (preprint), www.cse.uconn.edu/~tpeters.

D. Blackmore, M. C. Leu and F. Shih, Analysis and modelling of deformed swept volumes, CAD 26 (1994), 315–326.

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