Geometry & Topology

Discrete conformal maps and ideal hyperbolic polyhedra

Alexander I Bobenko, Ulrich Pinkall, and Boris A Springborn

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We establish a connection between two previously unrelated topics: a particular discrete version of conformal geometry for triangulated surfaces, and the geometry of ideal polyhedra in hyperbolic three-space. Two triangulated surfaces are considered discretely conformally equivalent if the edge lengths are related by scale factors associated with the vertices. This simple definition leads to a surprisingly rich theory featuring Möbius invariance, the definition of discrete conformal maps as circumcircle-preserving piecewise projective maps, and two variational principles. We show how literally the same theory can be reinterpreted to address the problem of constructing an ideal hyperbolic polyhedron with prescribed intrinsic metric. This synthesis enables us to derive a companion theory of discrete conformal maps for hyperbolic triangulations. It also shows how the definitions of discrete conformality considered here are closely related to the established definition of discrete conformality in terms of circle packings.

Article information

Geom. Topol., Volume 19, Number 4 (2015), 2155-2215.

Received: 16 September 2013
Revised: 4 August 2014
Accepted: 12 October 2014
First available in Project Euclid: 16 November 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 52C26: Circle packings and discrete conformal geometry
Secondary: 52B10: Three-dimensional polytopes 57M50: Geometric structures on low-dimensional manifolds

discrete conformal geometry polyhedron hyperbolic geometry


Bobenko, Alexander I; Pinkall, Ulrich; Springborn, Boris A. Discrete conformal maps and ideal hyperbolic polyhedra. Geom. Topol. 19 (2015), no. 4, 2155--2215. doi:10.2140/gt.2015.19.2155.

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  • A D Alexandrov, Convex polyhedra, Springer, Berlin (2005)
  • E M Andreev, On convex polyhedra in Lobačevskiĭ spaces, Math. USSR-Sb. 10 (1970) 413–440
  • E M Andreev, On convex polyhedra of finite volume in Lobačevskiĭ space, Math. USSR-Sb. 12 (1970) 255–259
  • A I Bobenko, T Hoffmann, B A Springborn, Minimal surfaces from circle patterns: geometry from combinatorics, Ann. of Math. 164 (2006) 231–264
  • A I Bobenko, I Izmestiev, Alexandrov's theorem, weighted Delaunay triangulations and mixed volumes, Ann. Inst. Fourier (Grenoble) 58 (2008) 447–505
  • A I Bobenko, B A Springborn, Variational principles for circle patterns and Koebe's theorem, Trans. Amer. Math. Soc. 356 (2004) 659–689
  • B Chow, F Luo, Combinatorial Ricci flows on surfaces, J. Differential Geom. 63 (2003) 97–129
  • Y Colin de Verdière, Un principe variationnel pour les empilements de cercles, Invent. Math. 104 (1991) 655–669
  • J Dai, X D Gu, F Luo, Variational principles for discrete surfaces, Adv. Lect. Math. 4, International Press, Somerville, MA (2008)
  • R J Duffin, Distributed and lumped networks, J. Math. Mech. 8 (1959) 793–826
  • D B A Epstein, R C Penner, Euclidean decompositions of noncompact hyperbolic manifolds, J. Differential Geom. 27 (1988) 67–80
  • V V Fock, Dual Teichmüller spaces
  • D Futer, F Guéritaud, From angled triangulations to hyperbolic structures, from: “Interactions between hyperbolic geometry, quantum topology and number theory”, (A Champanerkar, O Dasbach, E Kalfagianni, I Kofman, W Neumann, N Stoltzfus, editors), Contemp. Math. 541, Amer. Math. Soc. (2011) 159–182
  • M von Gagern, J Richter-Gebert, Hyperbolization of Euclidean ornaments, Electron. J. Combin. 16 (2009)
  • I M Gelfand, M M Kapranov, A V Zelevinsky, Discriminants, resultants and multidimensional determinants, Birkhäuser, Boston, MA (1994)
  • X D Gu, S-T Yau, Computational conformal geometry, Adv. Lect. Math. 3, International Press, Somerville, MA (2008)
  • Z-X He, O Schramm, Fixed points, Koebe uniformization and circle packings, Ann. of Math. 137 (1993) 369–406
  • R Kenyon, Lectures on dimers, from: “Statistical mechanics”, (S Sheffield, T Spencer, editors), IAS/Park City Math. Ser. 16, Amer. Math. Soc. (2009) 191–230
  • R Kenyon, A Okounkov, Planar dimers and Harnack curves, Duke Math. J. 131 (2006) 499–524
  • R Kenyon, A Okounkov, S Sheffield, Dimers and amoebae, Ann. of Math. 163 (2006) 1019–1056
  • P Koebe, Kontaktprobleme der konformen Abbildung, Ber. Sächs. Akad. Wiss. Leipzig, Math.-phys. Kl. 88 (1936) 141–164
  • G Leibon, Characterizing the Delaunay decompositions of compact hyperbolic surfaces, Geom. Topol. 6 (2002) 361–391
  • L Lewin, Polylogarithms and associated functions, North-Holland, New York (1981)
  • F Luo, Combinatorial Yamabe flow on surfaces, Commun. Contemp. Math. 6 (2004) 765–780
  • F Luo, Rigidity of polyhedral surfaces, from: “Fourth International Congress of Chinese Mathematicians”, (L Ji, K Liu, L Yang, S-T Yau, editors), AMS/IP Stud. Adv. Math. 48, Amer. Math. Soc. (2010) 201–217
  • G Mikhalkin, Amoebas of algebraic varieties and tropical geometry, from: “Different faces of geometry”, (S Donaldson, Y Eliashberg, M Gromov, editors), Int. Math. Ser. 3, Kluwer/Plenum, New York (2004) 257–300
  • J Milnor, Hyperbolic geometry: The first 150 years, Bull. Amer. Math. Soc. 6 (1982) 9–24
  • J Milnor, How to compute volume in hyperbolic space, from: “Collected papers, Vol. $1$”, Publish or Perish, Houston, TX (1994) 189–212
  • J Milnor, The Schläfli differential equality, from: “Collected papers, Vol. $1$”, Publish or Perish, Houston, TX (1994) 281–295
  • A Papadopoulos (editor), Handbook of Teichmüller theory, Vol. I, IRMA Lectures Math. Theor. Physics 11, Euro. Math. Soc., Zürich (2007)
  • M Passare, H Rullgård, Amoebas, Monge–Ampère measures and triangulations of the Newton polytope, Duke Math. J. 121 (2004) 481–507
  • R C Penner, The decorated Teichmüller space of punctured surfaces, Comm. Math. Phys. 113 (1987) 299–339
  • U Pinkall, K Polthier, Computing discrete minimal surfaces and their conjugates, Experiment. Math. 2 (1993) 15–36
  • I Rivin, Euclidean structures on simplicial surfaces and hyperbolic volume, Ann. of Math. 139 (1994) 553–580
  • I Rivin, Intrinsic geometry of convex ideal polyhedra in hyperbolic $3$–space, from: “Analysis, algebra, and computers in mathematical research”, (M Gyllenberg, editor), Lecture Notes Pure Appl. Math. 156, Dekker, New York (1994) 275–291
  • I Rivin, Combinatorial optimization in geometry, Adv. in Appl. Math. 31 (2003) 242–271
  • M Roček, R M Williams, The quantization of Regge calculus, Z. Phys. C 21 (1984) 371–381
  • B Rodin, D Sullivan, The convergence of circle packings to the Riemann mapping, J. Differential Geom. 26 (1987) 349–360
  • R K W Roeder, J H Hubbard, W D Dunbar, Andreev's theorem on hyperbolic polyhedra, Ann. Inst. Fourier $($Grenoble$)$ 57 (2007) 825–882
  • B Springborn, P Schröder, U Pinkall, Conformal equivalence of triangle meshes, ACM Trans. Graph. 27 (2008) 11
  • K Stephenson, Introduction to circle packing, Cambridge Univ. Press (2005)
  • W P Thurston, The geometry and topology of three-manifolds, lecture notes (1979) Available at \setbox0\makeatletter\@url {\unhbox0
  • W P Thurston, Minimal stretch maps between hyperbolic surfaces (1998)
  • M Troyanov, Les surfaces euclidiennes à singularités coniques, Enseign. Math. 32 (1986) 79–94