Ideal Hyperbolic Polyhedra and Discrete Uniformization
We provide a constructive, variational proof of Rivin’s realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem for spheres. The same variational method is also used to prove a discrete uniformization theorem of Gu...
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| Published in | Discrete & computational geometry Vol. 64; no. 1; pp. 63 - 108 |
|---|---|
| Main Author | |
| Format | Journal Article |
| Language | English |
| Published |
New York
Springer US
01.07.2020
Springer Nature B.V |
| Subjects | |
| Online Access | Get full text |
| ISSN | 0179-5376 1432-0444 |
| DOI | 10.1007/s00454-019-00132-8 |
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| Abstract | We provide a constructive, variational proof of Rivin’s realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem for spheres. The same variational method is also used to prove a discrete uniformization theorem of Gu et al. and a corresponding polyhedral realization result of Fillastre. The variational principles involve twice continuously differentiable functions on the decorated Teichmüller spaces
T
~
g
,
n
of punctured surfaces, which are analytic in each Penner cell, convex on each fiber over
T
g
,
n
, and invariant under the action of the mapping class group. |
|---|---|
| AbstractList | We provide a constructive, variational proof of Rivin’s realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem for spheres. The same variational method is also used to prove a discrete uniformization theorem of Gu et al. and a corresponding polyhedral realization result of Fillastre. The variational principles involve twice continuously differentiable functions on the decorated Teichmüller spaces
T
~
g
,
n
of punctured surfaces, which are analytic in each Penner cell, convex on each fiber over
T
g
,
n
, and invariant under the action of the mapping class group. We provide a constructive, variational proof of Rivin’s realization theorem for ideal hyperbolic polyhedra with prescribed intrinsic metric, which is equivalent to a discrete uniformization theorem for spheres. The same variational method is also used to prove a discrete uniformization theorem of Gu et al. and a corresponding polyhedral realization result of Fillastre. The variational principles involve twice continuously differentiable functions on the decorated Teichmüller spaces T~g,n of punctured surfaces, which are analytic in each Penner cell, convex on each fiber over Tg,n, and invariant under the action of the mapping class group. |
| Author | Springborn, Boris |
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| CitedBy_id | crossref_primary_10_1007_s00454_022_00458_w crossref_primary_10_1145_3478513_3480557 crossref_primary_10_1145_3658202 crossref_primary_10_1007_s10711_024_00901_7 crossref_primary_10_1093_imrn_rnae016 crossref_primary_10_1007_s42543_021_00036_8 crossref_primary_10_1016_j_aim_2023_109439 crossref_primary_10_1007_s10711_022_00746_y crossref_primary_10_1145_3550454_3555513 crossref_primary_10_1007_s00526_022_02248_x crossref_primary_10_1007_s10711_021_00621_2 crossref_primary_10_1007_s00526_022_02194_8 crossref_primary_10_1145_3618394 crossref_primary_10_1007_s00526_022_02184_w crossref_primary_10_1093_bioinformatics_btad383 crossref_primary_10_1007_s00454_023_00484_2 crossref_primary_10_1093_imrn_rnab270 |
| Cites_doi | 10.1016/S0304-3975(00)00248-6 10.1016/0166-8641(93)90032-9 10.5802/aif.2358 10.1142/S0219199704001501 10.2307/2944356 10.1017/CBO9780511530067 10.4171/075 10.4310/jdg/1203000270 10.24033/bsmf.2426 10.1090/tran/7483 10.1090/S0002-9939-00-05829-9 10.1007/s00454-007-9006-1 10.1007/s00454-008-9077-7 10.1007/978-3-662-50447-5 10.1007/s10711-008-9252-2 10.1007/s00454-016-9830-2 10.2140/gt.2015.19.2155 10.1007/BF01264924 10.4310/jdg/1531188190 10.4310/jdg/1527040872 10.1007/s00454-004-1102-x 10.2140/gtm.1998.1.511 10.1016/0167-8396(90)90011-F 10.4310/jdg/1214441650 10.1090/S0273-0979-1982-14958-8 10.1142/8685 10.1080/10586458.1993.10504266 10.1007/BF01223515 10.1142/S0218195995000118 10.4310/jdg/1253804352 10.2140/gt.2009.13.457 10.1007/s10711-019-00480-y 10.1080/10586458.2018.1477078 |
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| Keywords | 57M50 52B10 52C26 Decorated Teichmüller space Penner coordinates Horocycle Discrete conformal equivalence Triangulated surface |
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| References | Duffin (CR11) 1959; 8 Moroianu, Schlenker (CR25) 2009; 83 Penner (CR26) 1987; 113 Aurenhammer, Klein, Lee (CR3) 2013 Fillastre, Izmestiev (CR15) 2009; 13 Tillmann, Wong (CR37) 2016; 7 Sakuma, Weeks (CR32) 1995; 55 CR33 Rivin, Gyllenberg, Persson (CR31) 1994 Bobenko, Pinkall, Springborn (CR7) 2015; 19 Bao, Bonahon (CR4) 2002; 130 Bobenko, Dimitrov, Sechelmann (CR6) 2017; 57 Weeks (CR38) 1993; 52 Schlenker (CR34) 2005; 33 Rippa (CR30) 1990; 7 Milnor (CR24) 1982; 6 Alexandrov (CR2) 2005 Thurston, Rivin, Rourke, Series (CR36) 1998 Epstein, Penner (CR13) 1988; 27 Pinkall, Polthier (CR28) 1993; 2 Masur, Smillie (CR23) 1991; 134 Bobenko, Izmestiev (CR5) 2008; 58 Bobenko, Springborn (CR9) 2007; 38 Luo (CR22) 2004; 6 CR29 Bowers, Bowers, Pratt (CR10) 2019; 371 Springborn (CR35) 2008; 78 Fillastre (CR14) 2008; 134 Fortune (CR16) 1995; 5 Gu, Guo, Luo, Sun, Wu (CR17) 2018; 109 CR21 Akiyoshi (CR1) 2001; 129 Edelsbrunner (CR12) 2001 Penner (CR27) 2012 Bobenko, Sechelmann, Springborn, Bobenko (CR8) 2016 Gu, Luo, Sun, Wu (CR18) 2018; 109 Indermitte, Liebling, Troyanov, Clémençon (CR19) 2001; 263 Izmestiev (CR20) 2008; 40 AI Bobenko (132_CR8) 2016 J-M Schlenker (132_CR34) 2005; 33 S Moroianu (132_CR25) 2009; 83 RC Penner (132_CR26) 1987; 113 U Pinkall (132_CR28) 1993; 2 X Bao (132_CR4) 2002; 130 M Sakuma (132_CR32) 1995; 55 I Izmestiev (132_CR20) 2008; 40 AD Alexandrov (132_CR2) 2005 X Gu (132_CR17) 2018; 109 WP Thurston (132_CR36) 1998 F Fillastre (132_CR15) 2009; 13 H Edelsbrunner (132_CR12) 2001 AI Bobenko (132_CR7) 2015; 19 RJ Duffin (132_CR11) 1959; 8 I Rivin (132_CR31) 1994 H Masur (132_CR23) 1991; 134 132_CR21 AI Bobenko (132_CR6) 2017; 57 J Milnor (132_CR24) 1982; 6 F Aurenhammer (132_CR3) 2013 S Fortune (132_CR16) 1995; 5 F Luo (132_CR22) 2004; 6 132_CR29 AI Bobenko (132_CR9) 2007; 38 BA Springborn (132_CR35) 2008; 78 S Tillmann (132_CR37) 2016; 7 RC Penner (132_CR27) 2012 S Rippa (132_CR30) 1990; 7 AI Bobenko (132_CR5) 2008; 58 H Akiyoshi (132_CR1) 2001; 129 XD Gu (132_CR18) 2018; 109 DBA Epstein (132_CR13) 1988; 27 JC Bowers (132_CR10) 2019; 371 C Indermitte (132_CR19) 2001; 263 JR Weeks (132_CR38) 1993; 52 132_CR33 F Fillastre (132_CR14) 2008; 134 |
| References_xml | – volume: 263 start-page: 263 issue: 1–2 year: 2001 end-page: 274 ident: CR19 article-title: Voronoi diagrams on piecewise flat surfaces and an application to biological growth publication-title: Theoret. Comput. Sci. doi: 10.1016/S0304-3975(00)00248-6 – volume: 8 start-page: 793 year: 1959 end-page: 826 ident: CR11 article-title: Distributed and lumped networks publication-title: J. Math. Mech. – volume: 52 start-page: 127 issue: 2 year: 1993 end-page: 149 ident: CR38 article-title: Convex hulls and isometries of cusped hyperbolic -manifolds publication-title: Topology Appl. doi: 10.1016/0166-8641(93)90032-9 – volume: 58 start-page: 447 issue: 2 year: 2008 end-page: 505 ident: CR5 article-title: Alexandrov’s theorem, weighted Delaunay triangulations, and mixed volumes publication-title: Ann. Inst. Fourier (Grenoble) doi: 10.5802/aif.2358 – volume: 6 start-page: 765 issue: 5 year: 2004 end-page: 780 ident: CR22 article-title: Combinatorial Yamabe flow on surfaces publication-title: Commun. Contemp. Math. doi: 10.1142/S0219199704001501 – volume: 134 start-page: 455 issue: 3 year: 1991 end-page: 543 ident: CR23 article-title: Hausdorff dimension of sets of nonergodic measured foliations publication-title: Ann. Math. doi: 10.2307/2944356 – ident: CR33 – year: 2001 ident: CR12 publication-title: Geometry and Topology for Mesh Generation. Cambridge Monographs on Applied and Computational Mathematics doi: 10.1017/CBO9780511530067 – year: 2012 ident: CR27 publication-title: Decorated Teichmüller Theory. QGM Master Class Series doi: 10.4171/075 – volume: 78 start-page: 333 issue: 2 year: 2008 end-page: 367 ident: CR35 article-title: A variational principle for weighted Delaunay triangulations and hyperideal polyhedra publication-title: J. Differ. Geom. doi: 10.4310/jdg/1203000270 – ident: CR29 – volume: 130 start-page: 457 issue: 3 year: 2002 end-page: 491 ident: CR4 article-title: Hyperideal polyhedra in hyperbolic 3-space publication-title: Bull. Soc. Math. France doi: 10.24033/bsmf.2426 – volume: 371 start-page: 4215 issue: 6 year: 2019 end-page: 4249 ident: CR10 article-title: Rigidity of circle polyhedra in the 2-sphere and of hyperideal polyhedra in hyperbolic 3-space publication-title: Trans. Am. Math. Soc. doi: 10.1090/tran/7483 – volume: 129 start-page: 2431 issue: 8 year: 2001 end-page: 2439 ident: CR1 article-title: Finiteness of polyhedral decompositions of cusped hyperbolic manifolds obtained by the Epstein–Penner’s method publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-00-05829-9 – volume: 38 start-page: 740 issue: 4 year: 2007 end-page: 756 ident: CR9 article-title: A discrete Laplace–Beltrami operator for simplicial surfaces publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-007-9006-1 – volume: 40 start-page: 561 issue: 4 year: 2008 end-page: 585 ident: CR20 article-title: A variational proof of Alexandrov’s convex cap theorem publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-008-9077-7 – start-page: 1 year: 2016 end-page: 56 ident: CR8 article-title: Discrete conformal maps: boundary value problems, circle domains, Fuchsian and Schottky uniformization publication-title: Advances in Discrete Differential Geometry doi: 10.1007/978-3-662-50447-5 – volume: 134 start-page: 177 year: 2008 end-page: 196 ident: CR14 article-title: Polyhedral hyperbolic metrics on surfaces publication-title: Geom. Dedicata doi: 10.1007/s10711-008-9252-2 – ident: CR21 – volume: 57 start-page: 431 issue: 2 year: 2017 end-page: 469 ident: CR6 article-title: Discrete uniformization of polyhedral surfaces with non-positive curvature and branched covers over the sphere via hyper-ideal circle patterns publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-016-9830-2 – volume: 19 start-page: 2155 issue: 4 year: 2015 end-page: 2215 ident: CR7 article-title: Discrete conformal maps and ideal hyperbolic polyhedra publication-title: Geom. Topol. doi: 10.2140/gt.2015.19.2155 – volume: 55 start-page: 115 issue: 2 year: 1995 end-page: 123 ident: CR32 article-title: The generalized tilt formula publication-title: Geom. Dedicata doi: 10.1007/BF01264924 – volume: 109 start-page: 431 issue: 3 year: 2018 end-page: 466 ident: CR17 article-title: A discrete uniformization theorem for polyhedral surfaces II publication-title: J. Differ. Geom. doi: 10.4310/jdg/1531188190 – volume: 109 start-page: 223 issue: 2 year: 2018 end-page: 256 ident: CR18 article-title: A discrete uniformization theorem for polyhedral surfaces publication-title: J. Differ. Geom. doi: 10.4310/jdg/1527040872 – volume: 33 start-page: 207 issue: 2 year: 2005 end-page: 221 ident: CR34 article-title: A rigidity criterion for non-convex polyhedra publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-004-1102-x – start-page: 511 year: 1998 end-page: 549 ident: CR36 article-title: Shapes of polyhedra and triangulations of the sphere publication-title: The Epstein Birthday Schrift. Geometry & Topology Monographs doi: 10.2140/gtm.1998.1.511 – volume: 7 start-page: 489 issue: 6 year: 1990 end-page: 497 ident: CR30 article-title: Minimal roughness property of the Delaunay triangulation publication-title: Comput. Aided Geom. Des. doi: 10.1016/0167-8396(90)90011-F – volume: 27 start-page: 67 issue: 1 year: 1988 end-page: 80 ident: CR13 article-title: Euclidean decompositions of noncompact hyperbolic manifolds publication-title: J. Differ. Geom. doi: 10.4310/jdg/1214441650 – start-page: 275 year: 1994 end-page: 291 ident: CR31 article-title: Intrinsic geometry of convex ideal polyhedra in hyperbolic -space publication-title: Analysis, Algebra, and Computers in Mathematical Research. Lecture Notes in Pure and Appl. Math., – volume: 7 start-page: 237 issue: 1 year: 2016 end-page: 255 ident: CR37 article-title: An algorithm for the Euclidean cell decomposition of a cusped strictly convex projective surface publication-title: J. Comput. Geom. – volume: 6 start-page: 9 issue: 1 year: 1982 end-page: 24 ident: CR24 article-title: Hyperbolic geometry: the first 150 years publication-title: Bull. Am. Math. Soc. (N.S.) doi: 10.1090/S0273-0979-1982-14958-8 – year: 2013 ident: CR3 publication-title: Voronoi Diagrams and Delaunay Triangulations doi: 10.1142/8685 – volume: 2 start-page: 15 issue: 1 year: 1993 end-page: 36 ident: CR28 article-title: Computing discrete minimal surfaces and their conjugates publication-title: Exp. Math. doi: 10.1080/10586458.1993.10504266 – volume: 113 start-page: 299 issue: 2 year: 1987 end-page: 339 ident: CR26 article-title: The decorated Teichmüller space of punctured surfaces publication-title: Commun. Math. Phys. doi: 10.1007/BF01223515 – year: 2005 ident: CR2 publication-title: Convex Polyhedra. Springer Monographs in Mathematics – volume: 5 start-page: 193 issue: 1–2 year: 1995 end-page: 213 ident: CR16 article-title: Numerical stability of algorithms for 2D Delaunay triangulations publication-title: Int. J. Comput. Geom. Appl. doi: 10.1142/S0218195995000118 – volume: 83 start-page: 75 issue: 1 year: 2009 end-page: 129 ident: CR25 article-title: Quasi-Fuchsian manifolds with particles publication-title: J. Differ. Geom. doi: 10.4310/jdg/1253804352 – volume: 13 start-page: 457 issue: 1 year: 2009 end-page: 492 ident: CR15 article-title: Hyperbolic cusps with convex polyhedral boundary publication-title: Geom. Topol. doi: 10.2140/gt.2009.13.457 – volume: 371 start-page: 4215 issue: 6 year: 2019 ident: 132_CR10 publication-title: Trans. Am. Math. Soc. doi: 10.1090/tran/7483 – volume: 109 start-page: 431 issue: 3 year: 2018 ident: 132_CR17 publication-title: J. Differ. Geom. doi: 10.4310/jdg/1531188190 – volume-title: Geometry and Topology for Mesh Generation. Cambridge Monographs on Applied and Computational Mathematics year: 2001 ident: 132_CR12 doi: 10.1017/CBO9780511530067 – volume: 83 start-page: 75 issue: 1 year: 2009 ident: 132_CR25 publication-title: J. Differ. Geom. doi: 10.4310/jdg/1253804352 – volume: 263 start-page: 263 issue: 1–2 year: 2001 ident: 132_CR19 publication-title: Theoret. Comput. Sci. doi: 10.1016/S0304-3975(00)00248-6 – volume: 2 start-page: 15 issue: 1 year: 1993 ident: 132_CR28 publication-title: Exp. Math. doi: 10.1080/10586458.1993.10504266 – start-page: 511 volume-title: The Epstein Birthday Schrift. Geometry & Topology Monographs year: 1998 ident: 132_CR36 doi: 10.2140/gtm.1998.1.511 – volume: 5 start-page: 193 issue: 1–2 year: 1995 ident: 132_CR16 publication-title: Int. J. Comput. Geom. Appl. doi: 10.1142/S0218195995000118 – volume: 7 start-page: 237 issue: 1 year: 2016 ident: 132_CR37 publication-title: J. Comput. Geom. – volume-title: Convex Polyhedra. Springer Monographs in Mathematics year: 2005 ident: 132_CR2 – volume: 113 start-page: 299 issue: 2 year: 1987 ident: 132_CR26 publication-title: Commun. Math. Phys. doi: 10.1007/BF01223515 – start-page: 1 volume-title: Advances in Discrete Differential Geometry year: 2016 ident: 132_CR8 doi: 10.1007/978-3-662-50447-5 – volume: 13 start-page: 457 issue: 1 year: 2009 ident: 132_CR15 publication-title: Geom. Topol. doi: 10.2140/gt.2009.13.457 – volume: 109 start-page: 223 issue: 2 year: 2018 ident: 132_CR18 publication-title: J. Differ. Geom. doi: 10.4310/jdg/1527040872 – volume: 6 start-page: 9 issue: 1 year: 1982 ident: 132_CR24 publication-title: Bull. Am. Math. Soc. (N.S.) doi: 10.1090/S0273-0979-1982-14958-8 – volume-title: Decorated Teichmüller Theory. QGM Master Class Series year: 2012 ident: 132_CR27 doi: 10.4171/075 – volume-title: Voronoi Diagrams and Delaunay Triangulations year: 2013 ident: 132_CR3 doi: 10.1142/8685 – volume: 6 start-page: 765 issue: 5 year: 2004 ident: 132_CR22 publication-title: Commun. Contemp. Math. doi: 10.1142/S0219199704001501 – volume: 58 start-page: 447 issue: 2 year: 2008 ident: 132_CR5 publication-title: Ann. Inst. Fourier (Grenoble) doi: 10.5802/aif.2358 – volume: 8 start-page: 793 year: 1959 ident: 132_CR11 publication-title: J. Math. Mech. – ident: 132_CR29 doi: 10.1007/s10711-019-00480-y – volume: 7 start-page: 489 issue: 6 year: 1990 ident: 132_CR30 publication-title: Comput. Aided Geom. Des. doi: 10.1016/0167-8396(90)90011-F – ident: 132_CR21 doi: 10.1080/10586458.2018.1477078 – volume: 134 start-page: 455 issue: 3 year: 1991 ident: 132_CR23 publication-title: Ann. Math. doi: 10.2307/2944356 – volume: 55 start-page: 115 issue: 2 year: 1995 ident: 132_CR32 publication-title: Geom. Dedicata doi: 10.1007/BF01264924 – volume: 130 start-page: 457 issue: 3 year: 2002 ident: 132_CR4 publication-title: Bull. Soc. Math. France doi: 10.24033/bsmf.2426 – volume: 38 start-page: 740 issue: 4 year: 2007 ident: 132_CR9 publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-007-9006-1 – volume: 129 start-page: 2431 issue: 8 year: 2001 ident: 132_CR1 publication-title: Proc. Am. Math. Soc. doi: 10.1090/S0002-9939-00-05829-9 – start-page: 275 volume-title: Analysis, Algebra, and Computers in Mathematical Research. Lecture Notes in Pure and Appl. Math., year: 1994 ident: 132_CR31 – volume: 27 start-page: 67 issue: 1 year: 1988 ident: 132_CR13 publication-title: J. Differ. Geom. doi: 10.4310/jdg/1214441650 – volume: 52 start-page: 127 issue: 2 year: 1993 ident: 132_CR38 publication-title: Topology Appl. doi: 10.1016/0166-8641(93)90032-9 – volume: 57 start-page: 431 issue: 2 year: 2017 ident: 132_CR6 publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-016-9830-2 – volume: 134 start-page: 177 year: 2008 ident: 132_CR14 publication-title: Geom. Dedicata doi: 10.1007/s10711-008-9252-2 – volume: 78 start-page: 333 issue: 2 year: 2008 ident: 132_CR35 publication-title: J. Differ. Geom. doi: 10.4310/jdg/1203000270 – volume: 40 start-page: 561 issue: 4 year: 2008 ident: 132_CR20 publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-008-9077-7 – ident: 132_CR33 – volume: 19 start-page: 2155 issue: 4 year: 2015 ident: 132_CR7 publication-title: Geom. Topol. doi: 10.2140/gt.2015.19.2155 – volume: 33 start-page: 207 issue: 2 year: 2005 ident: 132_CR34 publication-title: Discrete Comput. Geom. doi: 10.1007/s00454-004-1102-x |
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| SubjectTerms | Combinatorics Computational Mathematics and Numerical Analysis Mapping Mathematical analysis Mathematics Mathematics and Statistics Polyhedra Theorems Variational principles |
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