We ask the question of which quantities one should measure on a discrete object such. While treating the material at an elementary level, the book also highlights many recent developments. From 3d consistency to zero curvature representations and b. Dec 16, 2008 rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. Recent progress in discrete differential geometry has lead, somewhat unexpectedly, to a better understanding of some fundamental structures lying in the basis of the classical differential geometry and of the theory of integrable systems. The integrability of the underlying equilibrium equations is proved by relating the geometry of the discrete shell membranes to discrete o surface theory. It reflects the recent progress in discrete differential geometry and contains many original results. We present the first steps of a procedure which discretizes surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. Ams grad stud math 98, providence 2008 currently the authoritive source for this branch of mathematics, where also most of the material presented here may be found. Pdf in this paper, we construct darboux transformations for two second order difference equations, and develop the associated crums theorems. It mainly deals with polygonal curves and polyhedral surfaces whose properties are analogous to continuous counterparts, where the smooth theory is established as limit of the discrete theory.
Bobenko, technische universitat berlin, berlin, germany and yuri b. It is used in the study of computer graphics and topological combinatorics. Nets in quadrics special classes of discrete surfaces. Apr 18, 2005 recent progress in discrete differential geometry has lead, somewhat unexpectedly, to a better understanding of some fundamental structures lying in the basis of the classical differential geometry and of the theory of integrable systems. The goal of the oberwolfach seminar discrete differential geometry held.
Often such a discretization clarifies the structures of the smooth theory and. Suris graduate studies in mathematics volume 98 editorial board david cox chair steven g. The ba sic philosophy of discrete differential geometry is that. Several classes of surfaces in differential geometry were shown to possess integrable structures. Using cones of an integer lattice, we introduce tangent bundlelike structure on a collection of nsimplices naturally. Thurston 1980s developed koebes ideas of discrete complex analysis based on circle patterns. Discrete differential geometry of n simplices and protein.
The basis of our model is a lesserknown characterization of developable surfaces as manifolds that can be parameterized through orthogonal geodesics. Advances in discrete differential geometry by alexander i. Discrete geometry and combinatorial geometry are branches of geometry that study combinatorial properties and constructive methods of discrete geometric objects. Discrete differential geometry develops discrete equivalents of notions and methods of classical differential geometry the latter appears as limit of the refinement of the discretization basic structures of ddg related to the theory of integrable systems a. Calculus, of differential, yet readily discretizable computational foundations is a crucial ingredient for numerical. Towards a unified theory of discrete surfaces with constant mean curvature, in. What is discrete differential geometry integrability from discrete to smooth structure of this book how to read this book acknowledgements chapter 1. We discuss a new geometric approach to discrete integrability coming from discrete differential geometry. Cse891 discrete differential geometry 3 a bit of history geometry is the key. Discrete differential geometry integrable structure alexander i. It is observed that koenigs nets come in pairs, since the discrete conjugate nets r and appear on equal footing.
Integrable structure in discrete shell membrane theory. Wilczynskis formalism, which was adopted by bol in the first two volumes of his monograph projektive differentialgeometrie,14, turns out to be custommade in connection with not only the isolation of integrable structure but also the development of a canonical discrete analogue of projective differential geometry within the field of. The subject focuses on the combinatorial properties of these. Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. A glimpse into discrete differential geometry geometry collective. Geometry ii discrete di erential geometry tu berlin. This paper proposes a novel discrete differential geometry of nsimplices. This is one of the first books on a newly emerging field of discrete differential geometry and an excellent way to access this exciting area. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. Surface theory in discrete projective differential. Pdf on the lagrangian structure of integrable hierarchies.
It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It was originally developed for protein structure analysis. The overarching themes introduced here, convergence and structure preservation, make repeated appearances throughout the entire volume. Pdf discrete crums theorems and integrable lattice equations. This is one of the first books on a newly emerging field of discrete differential. Discrete differential geometry is the study of discrete equivalents of the geometric notions and methods of classical differential geometry. On the integrability of infinitesimal and finite deformations of polyhedral surfaces. It is used in the study of computer graphics and topological combinatorics see also. The contribution of discrete differential geometry to. Important di erence equations related to integrable systems, special classes of surfaces. We establish connections with generalized barycentric coordinates and ninepoint centres and identify a discrete version of the. This algorithmically verifiable property implies analytical structures characteristic of integrability, such as the zerocurvature representation. Freeform architecture and discrete differential geometry.
Discrete geometry basic tool differential geometry metric, curvature, etc. Integrable structure 2008 advances in discrete differential geometry 2016 ams short course notes on ddg 2017. May 08, 2014 discrete koenigs nets were originally proposed by sauer and analysed in detail in 32,33 in connection with integrable discrete differential geometry. Graduate studies in mathematics publication year 2008. Bobenko and pinkall 4 considered discrete lax representations of isothermic surfaces and. For a given smooth geometry one can suggest many different discretizations. Yuri b suris an emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry.
From 3d consistency to zero curvature representations and backlund transformations 222 6. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. It should go without saying that this work is a major contribution to mathematics. We have applied the mathematical framework to analysis of. Whereas classical differential geometry investigates smooth geometric shapes such as surfaces, and discrete geometry studies geometric shapes with finite number of elements such as polyhedra, the discrete differential geometry aims at the development of. We propose a canonical frame in terms of which the associated projective gaussweingarten and gaussmainardicodazzi equations adopt compact forms. Discrete differential geometry graduate studies in. Geometry of boundary value problems for integrable 2d. An applied introduction siggraph 2005 course please note. Alternative analytic description of conjugate nets 1. Because many of the standard tools used in differential geometry have discrete combinatorial analogs, the discrete versions of forms or manifolds will be formally identical to and should partake of the same.
Schief, bobenko and hoffmann consider the rigidity of. Pdf discrete differential geometry of n simplices and. Secondary 51axx, 51bxx, 53axx, 37kxx, 39a12, 52c26. The goal of this book is to give a systematic presentation of current achievements in this field. On discrete differential geometry in twistor space. Pdf this paper proposes a novel discrete differential geometry of nsimplices. August 5, 2017 the paper the strong ring of simplicial complexes introduces a ring of geometric objects in which one can compute quantities like cohomologies faster. Discrete geodesic nets for modeling developable surfaces. The subject is simple topology or discrete differential geometry initiated in this paper. This first introductory text to discrete integrable systems introduces key notions of integrability from the vantage point of discrete systems, also making connections with the continuous theory where relevant. Springer this is the book on a newly emerging field of discrete differential geometry.
Destination page number search scope search text search scope search text. Discrete differential geometry integrable structure. Discrete differential geometry is an active mathematical terrain where. Integrable structure discrete differential geometry. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also from its applications in computer graphics, theoretical physics, architecture, and numerics. Laguerre geometry laguerre geometry is the geometry of oriented planes and oriented spheres in euclidean 3space. The application of discrete surfaces in architecture is discussed. On the other hand, it is addressed to specialists in geometry and mathematical physics.
A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Discrete differential geometry, integrable structure. Krantz rafe mazzeo martin scharlemann 2000 mathematics subject classification. Advances in discrete differential geometry springerlink. From discrete differential geometry to the classification of discrete. One may formalize the concept of integrable g gstructure in the generality of higher differential geometry, formalized in differential cohesion. Discrete differential geometry is an active mathematical terrain where differential geometry and discrete geometry meet and interact. It also provides a short survey of recent developments in digital geometry processing and discrete differential geometry. The goal is to understand graphs on a geometric level and investigate discrete analogues of structures which are known in differential geometry. Most questions in discrete geometry involve finite or discrete sets of basic geometric objects, such as points, lines, planes, circles, spheres, polygons, and so forth.
Billiards in confocal quadrics as a plurilagrangian system. Alexandrov starting 1950s metric geometry of discrete surfaces. Classical differential geometry discretization principles. We present a discrete theory for modeling developable surfaces as quadrilateral meshes satisfying simple angle constraints. Unlike previous works, we consider connection between spacefilling nsimplices. See also there at differential cohesion gstructure. Projective differential geometry see 1,2 and references therein has been demonstrated to be a rich source of surface geometries which are governed by integrable partial differential equations 3,4.
The 2006 course notes, above, include many important corrections as well as valuable additional chapters. Pdf discrete crums theorems and integrable lattice. Discrete di erential geometry, integrable structure. Suris, technische universitat munchen, garching bei munchen, germany. Approximation of smooth surfaces by polyhedral surfaces. They are related to solitons in some partial differential equations. Surface theory in discrete projective differential geometry. One of the main goals of this book is to reveal this integrable structure of discrete differential geometry. In this context, the appropriate formalism has proved to be that of the american school founded by wilczynski who, in fact, initiated projective differential geometry 57.
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