Skip to main content

Secondary Polytope and Secondary Power Diagram

Computational Mathematics and Mathematical Physics volume 59, pages 1965–1981 (2019)Cite this article

Abstract

An ingenious construction of Gel’fand, Kapranov, and Zelevinsky [5] geometrizes the triangulations of a point configuration, such that all coherent triangulations form a convex polytope, the so-called secondary polytope. The secondary polytope can be treated as a weighted Delaunay triangulation in the space of all possible coherent triangulations. Naturally, it should have a dual diagram. In this work, we explicitly construct the secondary power diagram, which is the power diagram of the space of all possible power diagrams with nonempty boundary cells. Secondary power diagram gives an alternative proof for the classical secondary polytope theorem based on Alexandrov theorem. Furthermore, secondary power diagram theory shows one can transform a nondegenerated coherent triangulation to another nondegenerated coherent triangulation by a sequence of bistellar modifications, such that all the intermediate triangulations are nondegenerated and coherent. As an application of this theory, we propose an algorithm to triangulate a special class of 3d nonconvex polyhedra without using additional vertices. We prove that this algorithm terminates in \(O({{n}^{3}})\) time.

This is a preview of subscription content, access via your institution.

Access options

Buy single article

Instant access to the full article PDF.

USD 39.95

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Fig. 1.
Fig. 2.
Fig. 3.
Fig. 4.
Fig. 5.
Fig. 6.
Fig. 7.
Fig. 8.
Fig. 9.
Fig. 10.

REFERENCES

  1. A. D. Alexandrov, Convex Polyhedra, Translated from the 1950 Russian edition by N. S. Dairbekov, S. S. Kutateladze, and A. B. Sossinsky (Springer-Verlag, Berlin, 2005).

  2. F. Aurenhammer, “Power diagrams: Properties, algorithms, and applications,” SIAM J. Comput. 16 (1), 78–96 (1987).

    MathSciNet  Article  Google Scholar 

  3. L. J. Billera, P. Filliman, and B. Sturmfels, “Constructions and complexity of secondary polytopes,” Adv. Math. 83 (2), 155–179 (1990).

    MathSciNet  Article  Google Scholar 

  4. E. F. D’Azevedo, “Optimal triangular mesh generation by coordinate transformation,” SIAM J. Sci. Stat. Comput. 12 (4), 755–786 (1991).

    MathSciNet  Article  Google Scholar 

  5. I. M. Gel’fand, M. M. Kapranov, and A. V. Zelevinsky, Discriminants, Resultants, and Multidimensional Determinants (Birkhäuser, 1994).

    Book  Google Scholar 

  6. X. Gu, F. Luo, J. Sun, and S.-T. Yau, “Variational principles for Minkowski type problems, discrete optimal transport, and discrete Monge–Ampere equations,” Asian J. Math. 20 (2), 383–398 (2016).

    MathSciNet  Article  Google Scholar 

  7. A. V. Zelevinsky, I. M. Gel’fand, and M. M. Kapranov, “Newton polyhedra of principal a-determinants,” Sov. Math. Dokl. 308, 20–23 (1989).

    MathSciNet  Google Scholar 

  8. I. Izmestiev, S. Klee, and I. Novik, “Simplicial moves on balanced complexes,” Adv. Math. 320, 82–114 (2017).

    MathSciNet  Article  Google Scholar 

  9. F. Santos, J. A. De Loera, and J. Rambau, Triangulations, Structures for Algorithms, and Applications (Springer-Verlag, Berlin, 2010).

    MATH  Google Scholar 

  10. B. Joe, “Construction of three-dimensional Delaunay triangulations using local transformations,” Comput.-Aided Geom. Des. 8 (2), 123–142 (1991).

    MathSciNet  Article  Google Scholar 

  11. J. Ruppert and R. Seidel, “On the difficulty of triangulating three-dimensional nonconvex polyhedra,” Discrete Comput. Geom. 7, 227–253 (1992).

    MathSciNet  Article  Google Scholar 

  12. H. Si, “On monotone sequences of directed flips, triangulations of polyhedra, and structural properties of a directed flip graph” (2018). http://arxiv.org/abs/1809.09701 arXiv:1809.09701

Download references

Author information

Authors and Affiliations

  1. DUT-RU ISE, Dalian University of Technology, 116620, Dalian, China

    Na Lei

  2. School of Software Technology, Dalian University of Technology, 116620, Dalian, China

    Wei Chen

  3. Key Laboratory for Ubiquitous Network and Service Software of Liaoning Province, 116620, Dalian, China

    Zhongxuan Luo

  4. Weierstrass Institute for Applied Analysis and Stochastics, 10117, Berlin, Germany

    Hang Si

  5. Department of Computer Science, Stony Brook University, NY 11794, Stony Brook, USA

    Xianfeng Gu

Authors
  1. Na Lei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Wei Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Zhongxuan Luo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Hang Si
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Xianfeng Gu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Wei Chen.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Na Lei, Chen, W., Luo, Z. et al. Secondary Polytope and Secondary Power Diagram. Comput. Math. and Math. Phys. 59, 1965–1981 (2019). https://doi.org/10.1134/S0965542519120121

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1134/S0965542519120121

Keywords:

  • upper envelope
  • convex hull
  • power diagram
  • weighted Delaunay triangulation
  • secondary polytope