Alexander I Bobenko Editor Advances in Discrete Differential Geometry Advances in Discrete Differential Geometry Alexander I Bobenko Editor Advances in Discrete Differential Geometry Editor Alexander I Bobenko Institut für Mathematik Technische Universität Berlin Berlin Germany ISBN 978-3-662-50446-8 DOI 10.1007/978-3-662-50447-5 ISBN 978-3-662-50447-5 (eBook) Library of Congress Control Number: 2016939574 © The Editor(s) (if applicable) and The Author(s) 2016 This book is published open access Open Access This book is distributed under the terms of the Creative Commons AttributionNoncommercial 2.5 License (http://creativecommons.org/licenses/by-nc/2.5/) which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited The images or other third party material in this book are included in the work’s Creative Commons license, unless indicated otherwise in the credit line; if such material is not included in the work’s Creative Commons license and the respective action is not permitted by statutory regulation, users will need to obtain permission from the license holder to duplicate, adapt or reproduce the material This work is subject to copyright All commercial rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed The use of general descriptive names, registered names, trademarks, service marks, etc in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made Printed on acid-free paper This Springer imprint is published by Springer Nature The registered company is Springer-Verlag GmbH Berlin Heidelberg Preface In this book we take a closer look at discrete models in differential geometry and dynamical systems The curves used are polygonal, surfaces are made from triangles and quadrilaterals, and time runs discretely Nevertheless, one can hardly see the difference to the corresponding smooth curves, surfaces, and classical dynamical systems with continuous time This is the paradigm of structure-preserving discretizations The common idea is to find and investigate discrete models that exhibit properties and structures characteristic of the corresponding smooth geometric objects and dynamical processes These important and characteristic qualitative features should already be captured at the discrete level The current interest and advances in this field are to a large extent stimulated by its relevance for computer graphics, mathematical physics, architectural geometry, etc The book focuses on differential geometry and dynamical systems, on smooth and discrete theories, and on pure mathematics and its practical applications It demonstrates this interplay using a range of examples, which include discrete conformal mappings, discrete complex analysis, discrete curvatures and special surfaces, discrete integrable systems, special texture mappings in computer graphics, and freeform architecture It was written by specialists from the DFG Collaborative Research Center “Discretization in Geometry and Dynamics” The work involved in this book and other selected research projects pursued by the Center was recently documented in the film “The Discrete Charm of Geometry” by Ekaterina Eremenko Lastly, the book features a wealth of illustrations, revealing that this new branch of mathematics is both (literally) beautiful and useful In particular the cover illustration shows the discretely conformally parametrized surfaces of the inflated letters A and B from the recent educational animated film “conform!” by Alexander Bobenko and Charles Gunn At this place, we want to thank the Deutsche Forschungsgesellschaft for its ongoing support Berlin, Germany November 2015 Alexander I Bobenko v Contents Discrete Conformal Maps: Boundary Value Problems, Circle Domains, Fuchsian and Schottky Uniformization Alexander I Bobenko, Stefan Sechelmann and Boris Springborn Discrete Complex Analysis on Planar Quad-Graphs Alexander I Bobenko and Felix Günther 57 Approximation of Conformal Mappings Using Conformally Equivalent Triangular Lattices 133 Ulrike Bücking Numerical Methods for the Discrete Map Za 151 Folkmar Bornemann, Alexander Its, Sheehan Olver and Georg Wechslberger A Variational Principle for Cyclic Polygons with Prescribed Edge Lengths 177 Hana Kouřimská, Lara Skuppin and Boris Springborn Complex Line Bundles Over Simplicial Complexes and Their Applications 197 Felix Knöppel and Ulrich Pinkall Holomorphic Vector Fields and Quadratic Differentials on Planar Triangular Meshes 241 Wai Yeung Lam and Ulrich Pinkall Vertex Normals and Face Curvatures of Triangle Meshes 267 Xiang Sun, Caigui Jiang, Johannes Wallner and Helmut Pottmann S-Conical CMC Surfaces Towards a Unified Theory of Discrete Surfaces with Constant Mean Curvature 287 Alexander I Bobenko and Tim Hoffmann vii viii Contents Constructing Solutions to the Björling Problem for Isothermic Surfaces by Structure Preserving Discretization 309 Ulrike Bücking and Daniel Matthes On the Lagrangian Structure of Integrable Hierarchies 347 Yuri B Suris and Mats Vermeeren On the Variational Interpretation of the Discrete KP Equation 379 Raphael Boll, Matteo Petrera and Yuri B Suris Six Topics on Inscribable Polytopes 407 Arnau Padrol and Günter M Ziegler DGD Gallery: Storage, Sharing, and Publication of Digital Research Data 421 Michael Joswig, Milan Mehner, Stefan Sechelmann, Jan Techter and Alexander I Bobenko Contributors Alexander I Bobenko Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Raphael Boll Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Folkmar Bornemann Zentrum Mathematik – M3, Technische Universität München, Garching bei München, Germany Ulrike Bücking Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Felix Günther Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Tim Hoffmann Zentrum Mathematik – M10, Technische Universität München, Garching bei München, Germany Alexander Its Department of Mathematical Sciences, Indiana University–Purdue University, Indianapolis, IN, USA Caigui Jiang King Abdullah Univ of Science and Technology, Thuwal, Saudi Arabia Michael Joswig Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Felix Knöppel Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Hana Kouřimská Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Wai Yeung Lam Technische Universität Berlin, Inst Für Mathematik, Berlin, Germany ix x Contributors Daniel Matthes Zentrum Mathematik – M8, Technische Universität München, Garching bei München, Germany Milan Mehner Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Sheehan Olver School of Mathematics and Statistics, The University of Sydney, Sydney, Australia Arnau Padrol Institut de Mathématiques de Jussieu - Paris Rive Gauche, UPMC Univ Paris 06, Paris Cedex 05, France Matteo Petrera Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Ulrich Pinkall Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Helmut Pottmann Vienna University of Technology, Wien, Austria Stefan Sechelmann Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Lara Skuppin Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Boris Springborn Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Xiang Sun King Abdullah Univ of Science and Technology, Thuwal, Saudi Arabia Yuri B Suris Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Jan Techter Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Mats Vermeeren Inst für Mathematik, Technische Universität Berlin, Berlin, Germany Johannes Wallner Graz University of Technology, Graz, Austria Georg Wechslberger Zentrum Mathematik – M3, Technische Universität München, Garching bei München, Germany Günter M Ziegler Inst für Mathematik, Freie Universität Berlin, Berlin, Germany Discrete Conformal Maps: Boundary Value Problems, Circle Domains, Fuchsian and Schottky Uniformization Alexander I Bobenko, Stefan Sechelmann and Boris Springborn Abstract We discuss several extensions and applications of the theory of discretely conformally equivalent triangle meshes (two meshes are considered conformally equivalent if corresponding edge lengths are related by scale factors attached to the vertices) We extend the fundamental definitions and variational principles from triangulations to polyhedral surfaces with cyclic faces The case of quadrilateral meshes is equivalent to the cross ratio system, which provides a link to the theory of integrable systems The extension to cyclic polygons also brings discrete conformal maps to circle domains within the scope of the theory We provide results of numerical experiments suggesting that discrete conformal maps converge to smooth conformal maps, with convergence rates depending on the mesh quality We consider the Fuchsian uniformization of Riemann surfaces represented in different forms: as immersed surfaces in R3 , as hyperelliptic curves, and as CP1 modulo a classical Schottky group, i.e., we convert Schottky to Fuchsian uniformization Extended examples also demonstrate a geometric characterization of hyperelliptic surfaces due to Schmutz Schaller Introduction Not one, but several sensible definitions of discrete holomorphic functions and discrete conformal maps are known today The oldest approach, which goes back to the early finite element literature, is to discretize the Cauchy–Riemann equaA.I Bobenko · S Sechelmann · B Springborn (B) Inst für Mathematik, Technische Universität Berlin, Straße des 17 Juni 136, 10623 Berlin, Germany e-mail: boris.springborn@tu-berlin.de A.I Bobenko e-mail: bobenko@math.tu-berlin.de S Sechelmann e-mail: sechel@math.tu-berlin.de c The Author(s) 2016 A.I Bobenko (ed.), Advances in Discrete Differential Geometry, DOI 10.1007/978-3-662-50447-5_1 .. .Advances in Discrete Differential Geometry Alexander I Bobenko Editor Advances in Discrete Differential Geometry Editor Alexander I Bobenko Institut für Mathematik Technische... from the respective cosine rules −α3 sin βi sin π−α1 −α 2 sin α2 sin α3 sin βi sin α1 +α2 2+α3 −π i = sin2 sin α2 sin α3 sinh2 i = (hyperbolic), (spherical) In both cases, expand the fraction on... Principles for Discrete Conformal Maps 3.1 Discrete Conformal Mapping Problems We will consider the following discrete conformal mapping problems (The notation ( , )g was introduced in Definition