Advances in discrete differential geometry

A book on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics.

Gespeichert in:
Bibliographische Detailangaben
1. Verfasser: Bobenko, Alexander I
Format: E-Book Buch
Sprache:Englisch
Veröffentlicht: Cham Springer Open 2016
Springer Nature
Springer Berlin / Heidelberg
Springer
Ausgabe:1
Schlagworte:
Differential Geometry
Discrete geometry
Geometry, Differential
Mathematics
Mathematics and Science
Nonfiction
ISBN:9783662504468, 3662504464, 3662504472, 9783662504475
Online-Zugang:Volltext
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Inhaltsangabe:
  • 2.3 Decomposing Velocity Fields into Fields Generated by Vortex Filaments -- 2.4 Close-To-Conformal Deformations of Volumes -- 3 Discrete Vector Bundles with Connection -- 4 Monodromy---A Discrete Analogue of Kobayashi's Theorem -- 5 Discrete Line Bundles---The Abelian Case -- 6 Discrete Connection Forms -- 7 Curvature---A Discrete Analogue of Weil's Theorem -- 8 The Index Formula for Hermitian Line Bundles -- 9 Piecewise-Smooth Vector Bundles over Simplicial Complexes -- 10 The Associated Piecewise-Smooth Hermitian Line Bundle -- 11 Finite Elements for Hermitian Line Bundles with Curvature -- 12 Discrete Energies on Surfaces---An Example -- References -- Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes -- 1 Introduction -- 2 Discrete Conformality -- 2.1 Conformal Equivalence -- 2.2 Circle Patterns -- 3 Infinitesimal Deformations and Linear Conformal Theory -- 3.1 Infinitesimal Deformations of a Triangle -- 3.2 Harmonic Functions with Respect to the Cotangent Laplacian -- 4 Holomorphic Quadratic Differentials -- 5 Conformal Deformations in Terms of End(mathbbC2) -- 6 Weierstrass Representation of Discrete Minimal Surfaces -- References -- Vertex Normals and Face Curvatures of Triangle Meshes -- 1 Introduction -- 2 Smooth Line Congruences -- 3 Discrete Normal Congruences -- 4 Curvatures of Faces of Triangle Meshes -- 5 Results and Discussion -- References -- S-Conical CMC Surfaces. Towards a Unified Theory of Discrete Surfaces with Constant Mean Curvature -- 1 Introduction -- 2 Conical Nets -- 3 Curvatures of Conical Nets via Steiner's Formula -- 4 Dual Quadrilaterals and Koenigs Nets -- 5 Conical Nets with Constant Mean Curvature and S-Conical Nets -- 6 S-Isothermic Nets -- 7 S-Conical Nets as S-Isothermic Nets -- 8 S-Conical Nets with Constant Mean Curvature -- 9 Delaunay Nets -- References
  • Intro -- Preface -- Contents -- Contributors -- Discrete Conformal Maps: Boundary Value Problems, Circle Domains, Fuchsian and Schottky Uniformization -- 1 Introduction -- 2 Discrete Conformal Equivalence of Cyclic Polyhedral Surfaces -- 2.1 Cyclic Polyhedral Surfaces -- 2.2 Notation -- 2.3 Discrete Metrics -- 2.4 Discrete Conformal Equivalence -- 2.5 Triangulations: Characterization by Length Cross-Ratios -- 2.6 Triangulations: Reconstructing Lengths from Length Cross-Ratios -- 2.7 Bipartite Graphs: Characterization by Length Multi-Ratios -- 2.8 Quadrangulations: The Cross-Ratio System on Quad-Graphs -- 3 Variational Principles for Discrete Conformal Maps -- 3.1 Discrete Conformal Mapping Problems -- 3.2 Analytic Formulation of the Mapping Problems -- 3.3 Variational Principles -- 3.4 The Triangle Functions -- 4 Conformal Maps of Cyclic Quadrangulations -- 4.1 Emerging Circle Patterns and a Necessary Condition -- 4.2 Riemann Maps with Cyclic Quadrilaterals -- 5 Multiply Connected Domains -- 5.1 Circle Domains -- 5.2 Special Slit Domains -- 6 Uniformization of Spheres -- 6.1 Uniformizing Quadrangulations of the Sphere -- 6.2 Using the Spherical Functional -- 6.3 Möbius Normalization -- 7 Uniformization of Tori -- 7.1 Immersed Tori -- 7.2 Elliptic Curves -- 7.3 Choosing Points on the Sphere -- 7.4 Numerical Experiments -- 7.5 Putting a Square Pattern on a Spherical Mesh -- 8 Uniformization of Surfaces of Higher Genus -- 8.1 Fundamental Polygons and Group Generators -- 8.2 From Schottky to Fuchsian Uniformization -- 8.3 Hyperelliptic Curves -- 8.4 Geometric Characterization of Hyperelliptic Surfaces -- 8.5 Example: Deforming a Hyperelliptic Surface -- 8.6 Example: Different Forms of the Same Genus-2 Surface -- References -- Discrete Complex Analysis on Planar Quad-Graphs -- 1 Introduction -- 2 Discrete Complex Analysis on Planar Quad-Graphs
  • 5 Universality -- 6 (i, j)-Scribability -- References -- DGD Gallery: Storage, Sharing, and Publication of Digital Research Data -- 1 Introduction -- 2 Comparison with Previous Work -- 3 Examples -- 3.1 Discrete S--Conical Catenoid and Helicoid -- 3.2 za Circle Pattern -- 3.3 Koebe Polyhedra -- 3.4 Lawson's Surface Uniformization -- 3.5 Tropical Grassmannian TropGr(2,6) -- 4 Architecture -- 4.1 What Is a Model? -- 4.2 Media Objects and Data Files -- 4.3 Versioning -- 4.4 Users -- 4.5 Submission Process -- 4.6 Publication and Licensing -- 5 Implementation -- 5.1 XML Based Backend and the XML Document Database BaseX -- 5.2 A Fail-Safe Release and Migration Process -- 5.3 A JavaScript Web Front End -- References
  • 2.1 Basic Definitions and Notation -- 2.2 Discrete Holomorphicity -- 2.3 Discrete Exterior Calculus -- 2.4 Discrete Laplacian -- 2.5 Discrete Green's Functions -- 2.6 Discrete Cauchy's Integral Formulae -- 3 Discrete Complex Analysis on Planar Parallelogram-Graphs -- 3.1 Preliminaries -- 3.2 Discrete Exponential Function -- 3.3 Asymptotics of the Discrete Green's Function -- 3.4 Asymptotics of Discrete Cauchy's Kernels -- 3.5 Integer Lattice -- References -- Approximation of Conformal Mappings Using Conformally Equivalent Triangular Lattices -- 1 Introduction -- 1.1 Convergence for Discrete Conformal PL-Maps on Triangular Lattices -- 1.2 Other Convergence Results for Discrete Conformal Maps -- 2 Some Characterizations of Associated Scale Factors of Discrete Conformal PL-Maps -- 3 Taylor Expansions -- 4 Existence of Discrete Conformal PL-Maps and Estimates -- References -- Numerical Methods for the Discrete Map Za -- 1 Introduction -- 2 Discrete Painlevé II Separatrix as a Boundary Value Problem -- 3 The Riemann--Hilbert Method -- 4 Lower Triangular Jump Matrices and Indices -- 5 RHPs as Integral Equations with Singular Kernels -- 6 A Well-Conditioned Spectral Method for Closed Contours -- 7 RHPs as Integral Equations with Nonsingular Kernels -- 8 A Modified Nyström Method -- 9 Conclusion -- References -- A Variational Principle for Cyclic Polygons with Prescribed Edge Lengths -- 1 Introduction -- 2 Euclidean Polygons. Proof of Theorem 1.1 -- 3 Spherical Polygons. Proof of Theorem 1.2 -- 4 Hyperbolic Polygons. Proof of Theorem 1.3 -- 5 Concluding Remarks on 1+1 Spacetime -- References -- Complex Line Bundles Over Simplicial Complexes and Their Applications -- 1 Introduction -- 2 Applications of Vector Bundles in Geometry Processing -- 2.1 Direction Fields on Surfaces -- 2.2 Stripe Patterns on Surfaces
  • Constructing Solutions to the Björling Problem for Isothermic Surfaces by Structure Preserving Discretization -- 1 Introduction -- 2 Smooth Isothermic Surfaces -- 2.1 Coordinates and Domains -- 2.2 Definition and Equations -- 2.3 Local Solution of the Björling Problem -- 3 Discrete Isothermic Surfaces -- 3.1 Coordinates and Domains -- 3.2 Definition of Discrete Isothermic Surfaces -- 3.3 The Discrete Björling Problem -- 3.4 Discrete Quantities and Basic Relations -- 3.5 Discrete Gauss-Codazzi System -- 4 The Abstract Convergence Result -- 4.1 Domains -- 4.2 Statement of the Approximation Result -- 4.3 Consistency -- 4.4 Stability -- 5 The Continuous Limit of Discrete Isothermic Surfaces -- 5.1 From Björling Data to Cauchy Data and Back -- 5.2 Main Result -- 6 Transformations -- 6.1 Christoffel Transformation -- 6.2 Darboux Transformation -- References -- On the Lagrangian Structure of Integrable Hierarchies -- 1 Introduction -- 2 Pluri-Lagrangian Systems -- 2.1 Definition -- 2.2 Approximation by Stepped Surfaces -- 2.3 Multi-time Euler-Lagrange Equations for Curves -- 2.4 Multi-time Euler-Lagrange Equations for Two-Dimensional Surfaces -- 3 Pluri-Lagrangian Structure of the Sine-Gordon Equation -- 4 The KdV Hierarchy -- 5 Pluri-Lagrangian Structure of PKdV Hierarchy -- 5.1 Variational Symmetries and the Pluri-Lagrangian Form -- 5.2 The Multi-time Euler-Lagrange Equations -- 6 Relation to Hamiltonian Formalism -- 7 Conclusion -- References -- On the Variational Interpretation of the Discrete KP Equation -- 1 Introduction -- 2 The Root Lattice Q(AN) -- 3 The dKP Equation on Q(AN) -- 4 The Cubic Lattice mathbbZN -- 5 The dKP Equation on mathbbZN -- 6 Conclusion -- References -- Six Topics on Inscribable Polytopes -- 1 Inscribability of 3-Polytopes -- 2 A Characterization in Higher Dimensions -- 3 Neighborly Polytopes -- 4 Universally Inscribable