Advances in Discrete Differential Geometry.

• Author: Bobenko, Alexander I.
• Place / Publishing House: Berlin, Heidelberg : : Springer Berlin / Heidelberg,, 2016.; ©2016.
• Year of Publication: 2016
• Edition: 1st ed.
• Language: English
• Physical Description: 1 online resource (441 pages)

Table of Contents

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.

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.

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.

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.

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.