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Discrete & Computational Geometry

, Volume 39, Issue 1–3, pp 339–388 | Cite as

Metric Combinatorics of Convex Polyhedra: Cut Loci and Nonoverlapping Unfoldings

  • Ezra Miller
  • Igor Pak
Article

Abstract

Let S be the boundary of a convex polytope of dimension d+1, or more generally let S be a convex polyhedral pseudomanifold. We prove that S has a polyhedral nonoverlapping unfolding into  \({\Bbb{R}}^{d}\) , so the metric space S is obtained from a closed (usually nonconvex) polyhedral ball in  \({\Bbb{R}}^{d}\) by identifying pairs of boundary faces isometrically. Our existence proof exploits geodesic flow away from a source point vS, which is the exponential map to S from the tangent space at v. We characterize the cut locus (the closure of the set of points in S with more than one shortest path to v) as a polyhedral complex in terms of Voronoi diagrams on facets. Analyzing infinitesimal expansion of the wavefront consisting of points at constant distance from v on S produces an algorithmic method for constructing Voronoi diagrams in each facet, and hence the unfolding of S. The algorithm, for which we provide pseudocode, solves the discrete geodesic problem. Its main construction generalizes the source unfolding for boundaries of three-polytopes into  \({\Bbb{R}}^{2}\) . We present conjectures concerning the number of shortest paths on the boundaries of convex polyhedra, and concerning continuous unfolding of convex polyhedra. We also comment on the intrinsic nonpolynomial complexity of nonconvex manifolds.

Keywords

Short Path Source Image Voronoi Diagram Discrete Comput Geom Convex Polyhedron 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Agarwal, P.K., Aronov, B., O’Rourke, J., Schevon, C.A.: Star unfolding of a polytope with applications. SIAM J. Comput. 26(6), 1689–1713 (1997) MATHCrossRefMathSciNetGoogle Scholar
  2. 2.
    Aggarwal, A., Guibas, L.J., Saxe, J., Shor, P.W.: A linear-time algorithm for computing the Voronoi diagram of a convex polygon. Discrete Comput. Geom. 4(6), 591–604 (1989) MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    Aleksandrov, A.D.: Vnutrennyaya geometriya vypuklykh poverkhnostey. Gostekhizdat, Moscow (1948) (in Russian). English translation: Selected Works. Intrinsic Geometry of Convex Surfaces, vol. 2. Chapman & Hall/CRC, Boca Raton (2005) Google Scholar
  4. 4.
    Aleksandrov, A.D.: Vypuklye mnogogranniki. Gostekhizdat, Moscow (1950) (in Russian). English translation: Convex Polyhedra, Springer, Berlin (2005) Google Scholar
  5. 5.
    Aronov, B., O’Rourke, J.: Nonoverlap of the star unfolding. Discrete Comput. Geom. 8(3), 219–250 (1992) MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Aurenhammer, F.: Voronoi diagrams—a survey of a fundamental geometric data structure. ACM Comput. Surv. 23, 345–405 (1991) CrossRefGoogle Scholar
  7. 7.
    Bern, M., Demaine, E.D., Eppstein, D., Kuo, E., Mantler, A., Snoeyink, J.: Ununfoldable polyhedra with convex faces. Comput. Geom. 24, 51–62 (2003) MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Bezdek, K., Connelly, R.: Pushing disks apart—the Kneser–Poulsen conjecture in the plane. J. Reine Angew. Math. 553, 221–236 (2002) MATHMathSciNetGoogle Scholar
  9. 9.
    Blum, L., Cucker, F., Shub, M., Smale, S.: Complexity and Real Computation. Springer, New York (1998) Google Scholar
  10. 10.
    Burago, Yu., Gromov, M., Perelman, A.D.: Alexandrov spaces with curvature bounded below. Russ. Math. Surv. 47(2), 1–58 (1992) MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Canny, J.F., Reif, J.H.: New lower bound techniques for robot motion planning problems. Proc. 28th IEEE FOCS, pp. 49–60 (1987) Google Scholar
  12. 12.
    Chazelle, B.: An optimal convex hull algorithm and new results on cuttings. In: Proc. 32nd IEEE FOCS, pp. 29–38 (1991) Google Scholar
  13. 13.
    Chen, J., Han, Y.: Shortest paths on a polyhedron. I. Computing shortest paths. Int. J. Comput. Geom. Appl. 6(2), 127–144 (1996) MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Connelly, R., Demaine, E.D., Rote, G.: Straightening polygonal arcs and convexifying polygonal cycles. Discrete Comput. Geom. 30(2), 205–239 (2003) MATHMathSciNetGoogle Scholar
  15. 15.
    Dijkstra, E.W.: A note on two problems in connexion with graphs. Numer. Math. 1, 269–271 (1959) MATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    Even, S., Goldreich, O.: The minimum-length generator sequence problem is NP-hard. J. Algorithms 2(3), 311–313 (1981) MATHCrossRefMathSciNetGoogle Scholar
  17. 17.
    Fortune, S.: Voronoi diagrams and Delaunay triangulations. In: Hwang, F., Du, D.Z. (eds.) Computing in Euclidean Geometry, pp. 225–265. World Scientific, Singapore (1995) Google Scholar
  18. 18.
    Goodman, J.E., O’Rourke, J. (eds.): Handbook of Discrete and Computational Geometry. CRC Press, Boca Raton (1997) MATHGoogle Scholar
  19. 19.
    Jerrum, M.R.: The complexity of finding minimum-length generator sequences. Theor. Comput. Sci. 36(2–3), 265–289 (1985) MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Kapoor, S.: An efficient computation of geodesic shortest paths. In: Proc. of the 31st ACM STOC, pp. 770–779 (1999) Google Scholar
  21. 21.
    Kobayashi, S.: On conjugate and cut loci. In: Global Differential Geometry, pp. 140–169. MAA, Washington (1989) Google Scholar
  22. 22.
    Kunze, R., Wolter, F.E., Rausch, T.: Geodesic Voronoi diagrams on parametric surfaces. In: Proc. Comput. Graphics Int., Hasselt-Diepenbeek, Belgium, pp. 230–237 (1997) Google Scholar
  23. 23.
    Lyusternik, L.A.: Geodesic Lines. The Shortest Paths on Surfaces. Gostekhizdat, Moscow (1940) (in Russian) Google Scholar
  24. 24.
    Mitchell, J.S.B.: Geometric shortest paths and network optimization. In: Handbook of Computational Geometry, pp. 633–701. North-Holland, Amsterdam (2000) Google Scholar
  25. 25.
    Mitchell, J.S.B., Sharir, M.: New results on shortest paths in three dimensions. In: Proc. 20th ACM Sympos. Comput. Geom., New York, pp. 124–133 (2004) Google Scholar
  26. 26.
    Mitchell, J.S.B., Mount, D.M., Papadimitriou, C.H.: The discrete geodesic problem. SIAM J. Comput. 16(4), 647–668 (1987) MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    Mount, D.M.: On finding shortest paths on convex polyhedra. Technical Report 1495, Dept. of Computer Science, University of Maryland, Baltimore, MD (1985) Google Scholar
  28. 28.
    O’Rourke, J.: Folding and unfolding in computational geometry. In: Discrete and Computational Geometry, Tokyo, 1998, pp. 258–266. Springer, Berlin (2000) Google Scholar
  29. 29.
    Papadopoulou, E., Lee, D.T.: A new approach for the geodesic Voronoi diagram of points in a simple polygon and other restricted polygonal domains. Algorithmica 20(4), 319–352 (1998) MATHCrossRefMathSciNetGoogle Scholar
  30. 30.
    Preparata, F.P., Shamos, M.I.: Computational Geometry. An Introduction. Texts and Monographs in Computer Science. Springer, New York (1985) Google Scholar
  31. 31.
    Schlickenrieder, W.: Nets of polyhedra. Diplomarbeit, TU Berlin, Berlin (1997) Google Scholar
  32. 32.
    Schreiber, Y., Sharir, M.: An efficient algorithm for shortest paths on a convex polytope in three dimensions, Preliminary version (see also the extended abstract in Proc. 22nd ACM Sympos. Comput. Geom., Sedona, AZ, 2006.) Google Scholar
  33. 33.
    Sharir, M., Schorr, A.: On shortest paths in polyhedral spaces. SIAM J. Comput. 15(1), 193–215 (1986) MATHCrossRefMathSciNetGoogle Scholar
  34. 34.
    Stone, D.A.: Geodesics in piecewise linear manifolds. Trans. Am. Math. Soc. 215, 1–44 (1976) MATHCrossRefGoogle Scholar
  35. 35.
    Surazhsky, V., Surazhsky, T., Kirsanov, D., Gortler, S.J., Hoppe, H.: Fast exact and approximate geodesics on meshes. ACM Trans. Graph. 24(3), 553–560 (2005) CrossRefGoogle Scholar
  36. 36.
    Tarasov, A.S.: Polyhedra that do not admit natural unfoldings. Russ. Math. Surv. 54(3), 656–657 (1999) MATHCrossRefMathSciNetGoogle Scholar
  37. 37.
    Volkov, J.A., Podgornova, E.G.: The cut locus of a polyhedral surface of positive curvature. Ukrain. Geom. Sb. 11, 15–25 (1971) (in Russian) MATHMathSciNetGoogle Scholar
  38. 38.
    Wolter, F.E.: Cut loci in bordered and unbordered Riemannian manifolds. Ph.D. thesis, TU Berlin, FB Mathematik, Berlin, Germany (1985) Google Scholar
  39. 39.
    Ziegler, G.M.: Lectures on Polytopes. Springer, New York (1995) MATHGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Department of MathematicsMITCambridgeUSA

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