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. 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. Discrete differential geometry integrable structure. Pdf a curvature theory for discrete surfaces based on mesh. Bobenko and suris cs177 2012 discrete differential geometry 2 differential and integral calculus study of invariants and symmetries. Current interest in discrete differential geometry derives not only from its importance in pure mathematics but also. Discrete differential geometry oberwolfach seminars 38.
On the other hand, it is addressed to specialists in geometry and mathematical physics. Towards a unified theory of discrete surfaces with constant mean curvature, in. Advances in discrete differential geometry springerlink. The basis of our model is a lesserknown characterization of developable surfaces as manifolds that can be parameterized through orthogonal geodesics. Discrete differential geometry is an active mathematical terrain where differential geometry and discrete geometry meet and interact. Discrete exterior calculus build your own dec at home. An emerging field of discrete differential geometry aims at the development of discrete equivalents of notions and methods of classical differential geometry. 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. Geometry of multilayer freeform structures for architecture. Roth theoretical kinematics dover publications 1990 t. As the twistor viewpoint relies on easy switching between the natural objects of their respective spaces via equivalences, certain constructions in this paper are illustrated by a triptych of equivalent diagrams in the 4sphere. They are a basic concept in the integrable system viewpoint of discrete differential geometry bobenko and suris 2005. Bobenko and tim hoffmann 67 discrete hashimoto surfaces and a doubly discrete smokering flow by tim hoffmann 95 the discrete greens function by yuri b.
It is used in the study of computer graphics and topological combinatorics see also. Integrable structure american mathematical society 2008 r. First book on a newly emerging field of discrete differential geometry. Pdf we consider a general theory of curvatures of discrete surfaces.
On discrete differential geometry in twistor space. 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. This article discusses the beautiful tale of how discrete differential geometry is linked to modern approaches to computational design for architecture, as well as fabrication and rationalization of freeform designs. 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. It surveys the fascinating connections between discrete models in differential geometry and complex analysis, integrable systems and applications in computer graphics. It also provides a short survey of recent developments in digital geometry processing and discrete differential geometry. Magic happens muchmorerobustmuch more robust is there a recipe. Dec 16, 2008 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. Short course on discrete differential geometry san diego. Classical differential geometry discretization principles. Conjugacyisa 3systeminddimensionalprojectivespace d 3, and circularity is a 3system in ddimensional mobius geometry.
Suris 2008, discrete differential geometry, american mathematical society, isbn 9780821847008 this differential geometry related article is a stub. Hoffmann discrete differential geometry of curves and surfaces faculty of mathematics, kyushu university 2009. Discrete curvature line parametrization in lie, mobius and laguerre geometries 115 3. Discrete differential geometry american mathematical society. Discrete differential geometry ddg is a new and active mathematical terrain where differential geometry providing the classical theory of smooth manifolds interacts with discrete geometry concerned with polytopes, simplicial complexes, etc. It provides discrete equivalents of the geometric notions and methods of differential geometry, such as notions of curvature and integrability for polyhedral surfaces. Advances in discrete differential geometry alexander i. Thus, discrete differential geometry in twistor space generalizes the theory of bobenko and suris for the lie quadric. Discrete differential geometry integrable structure alexander i.
Geometry ii discrete di erential geometry alexander i. The latter appears as a limit of a refinement of the discretization. Bottema topics in elementary geometry springer 2008 o. Suris supported by the dfg forschergruppe polyhedral surfaces and the dfg research center matheon mathematics for key technologies in berlin. On organizing principles of discrete differential geometry. More kinds of lax pairs are found for some newly appeared discrete integrable equations, including the h1, the special h3 and the q1 models in the adler bobenko suris list and the closely. Curvatures of discrete curves and surfaces 5 curves of finite total curvature by john m. Apr 18, 2005 a new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. From discrete differential geometry to the classification of discrete. Cs177 2012 discrete differential geometry 9 boys surface, oberwolfach is it much better. This is one of the first books on a newly emerging field of discrete differential geometry and an. Discrete asymptotic nets in plucker line geometry 118 3. Billiards in confocal quadrics as a plurilagrangian system.
The idea of discretizing smooth geometric notions forms the basis for fields such as discrete differential geometry and discrete exterior calculus 1, 5, 6. A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. We present a discrete theory for modeling developable surfaces as quadrilateral meshes satisfying simple angle constraints. Discrete geodesic nets for modeling developable surfaces. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes.
Discrete di erential geometry, integrable structure. The overarching themes introduced here, convergence and structure preservation, make repeated appearances throughout the entire volume. Most of the work relevant to the present study concerns quadrilateral meshes with planar faces, which discretize socalled conjugate curve networks on surfaces sauer 1970. It reflects the recent progress in discrete differential geometry and contains many original results. Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. He is an author with yuri suris of the book discrete differential geometry, editor of several books on this topic and a coorganizer of the regular conference discrete differential geometry in oberwolfach. The contribution of discrete differential geometry to.
We ask the question of which quantities one should measure on a discrete object such. It is a textbook on discrete differential geometry and integrable systems suitable for a one semester graduate course. He is an author with yuri suris of the book discrete differential geometry, editor of several books on. Geometry of spheres alexander bobenko technical university berlin lms durham symposium methods of integrable systems in geometry, august 1121, 2006 dfg research unit 565 polyhedral surfaces alexander bobenko on organizing principles of ddg. Discrete differential geometry graduate studies in. 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 discrete. Based on the lecture notes of geometry 2 summer semester 2014 tu berlin.
Suris american mathematical society providence, rhode island graduate studies. Bobenko and suris, 2008 i connections to integrable systems i applications in physics. Freeform architecture and discrete differential geometry. Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. Cse891 discrete differential geometry 3 a bit of history geometry is the key. Current interest in discrete differential geometry derives not. Rather unexpectedly, the very basic structures of discrete differential geometry turn out to be related to the theory of integrable systems. The contribution of discrete differential geometry to contemporary architecture. Bobenko is a professor at the technische universitat berlin. From discrete differential geometry to the classification of discrete integrable systems. The contribution of discrete differential geometry to contemporary architecture helmut pottmann vienna university of technology, austria. Bobenko and suris cs177 2011 discrete differential geometry 2 differential and integral calculus study of invariants and symmetries.
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