The Experts below are selected from a list of 80328 Experts worldwide ranked by ideXlab platform
Alexander I. Bobenko - One of the best experts on this subject based on the ideXlab platform.
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Advances in Discrete Differential Geometry - Advances in Discrete Differential Geometry
2016Co-Authors: Alexander I. BobenkoAbstract:This is one of the first books on a newly emergingfield of discreteDifferential Geometry and an excellent way to access this excitingarea. It surveys the fascinating connections between discrete modelsin Differential Geometry and complex analysis, integrable systems andapplications in computer graphics. The authors take a closer look at discrete models in Differential Geometry and dynamical systems. Their curves are polygonal, surfaces are made from triangles and quadrilaterals, and time is discrete. Nevertheless, the difference between the corresponding smooth curves, surfaces and classicaldynamical systems with continuoustime canhardly be seen. This is the paradigm ofstructure-preservingdiscretizations. Current advances in this field are stimulated to alarge extent by its relevance for computer graphics and mathematicalphysics. This book is written by specialists working together on acommon research project. It is about Differential Geometry anddynamical systems, smooth and discrete theories, and on puremathematics and its practical applications. The interaction of thesefacets is demonstrated by concrete examples, including discreteconformal mappings, discrete complex analysis, discrete curvatures andspecial surfaces, discrete integrablesystems, conformal texturemappings in computer graphics, and free-form architecture. This richly illustrated book will convincereaders that this newbranch of mathematics is both beautiful and useful. It will appeal tograduate students and researchers in Differential Geometry, complexanalysis, mathematical physics, numerical methods, discrete Geometry,as well as computer graphics and Geometry processing.
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discrete Differential Geometry integrable structure
2008Co-Authors: Alexander I. Bobenko, Yuri B. SurisAbstract:Classical Differential Geometry Discretization principles. Multidimensional nets Discretization principles. Nets in quadrics Special classes of discrete surfaces Approximation Consistency as integrability Discrete complex analysis. Linear theory Discrete complex analysis. Integrable circle patterns Foundations Solutions of selected exercises Bibliography Notations Index.
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Discrete Differential Geometry. Consistency as integrability
arXiv: Differential Geometry, 2005Co-Authors: Alexander I. Bobenko, Yuri B. SurisAbstract:A new field of discrete Differential Geometry is presently emerging on the border between Differential and discrete Geometry. 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 equivalents of notions and methods of smooth surface theory. Current interest in this field derives not only from its importance in pure mathematics but also from its relevance for other fields like computer graphics. 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 this book is to give a systematic presentation of current achievements in this field.
Hirokazu Nishimura - One of the best experts on this subject based on the ideXlab platform.
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Axiomatic Differential Geometry III-1
arXiv: Differential Geometry, 2012Co-Authors: Hirokazu NishimuraAbstract:In this paper is proposed a kind of model theory for our axiomatic Differential Geometry. It is claimed that smooth manifolds, which have occupied the center stage in Differential Geometry, should be replaced by functors on the category of Weil algebras. Our model theory is geometrically natural and conceptually motivated, while the model theory of synthetic Differential Geometry is highly artificial and exquisitely technical.
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Axiomatic Differential Geometry II-1 Vector Fields
arXiv: Differential Geometry, 2012Co-Authors: Hirokazu NishimuraAbstract:In our previous paper entitled "Axiomatic Differential Geometry -towards model categories of Differential Geometry-, we have given a category-theoretic framework of Differential Geometry. As the first part of our series of papers concerned with Differential-geometric developments within the above axiomatic scheme, this paper is devoted to vector fields. The principal result is that the totality of vector fields on a microlinear and Weil exponential object forms a Lie algebra.
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Curvature in Synthetic Differential Geometry of Groupoids
arXiv: Differential Geometry, 2007Co-Authors: Hirokazu NishimuraAbstract:We study the fundamental properties of curvature in groupoids within the framework of synthetic Differential Geometry. As is usual in synthetic Differential Geometry, its combinatorial nature is emphasized. In particular, the classical Bianchi identity is deduced from its combinatorial one.
Zhong Shan - One of the best experts on this subject based on the ideXlab platform.
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Research on SVG based on Differential Geometry nonlinear control
Application of Electronic Technique, 2010Co-Authors: Zhong ShanAbstract:According to the nonlinear character of SVG mathematic model,nonlinear control based on theory of Differential Geometry was proposed,using the nonlinear system which was I/O linear by Differential Geometry transmission,the nonlinear system was turned to a new linear system,then a controller of the system is designed according to the method of the exponentially approaching rule of variable structure theory.Results verify:controlling reactive current of SVG based on Differential Geometry nonlinear control has feasibility and validity.
Shan Zhong - One of the best experts on this subject based on the ideXlab platform.
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A Control Method for SVG Based on Differential Geometry Nonlinear Control
Energy and Power Engineering, 2011Co-Authors: Qiyong Pan, Yihuai Wang, Shan ZhongAbstract:The control method for SVG is researched in this paper. Based on the working mechanism of SVG, the logic switch function is introduced to establish the dynamic mathematic model. A Differential Geometry variable control method is provided and the Differential Geometry linear theory is used to convert the nonlinear system to a linear system. Then based on the former work the control of SVG is devised. Finally, the control of SVG is simulated and the result shows the Differential Geometry nonlinear control is robust and stable comparative to the traditional PID control method and it is an effective to control the SVG.
George Shapiro - One of the best experts on this subject based on the ideXlab platform.
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On discrete Differential Geometry in twistor space
Journal of Geometry and Physics, 2013Co-Authors: George ShapiroAbstract:Abstract In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in S 4 to complex values of a generalized cross-ratio by considering S 4 as a real section of the complex Plucker quadric, realized as the space of two-spheres in S 4 . We develop the Geometry of the Plucker quadric by examining the novel contact properties of two-spheres in S 4 , generalizing classical Lie Geometry in S 3 . Discrete Differential Geometry aims to develop discrete equivalents of the geometric notions and methods of classical Differential Geometry. We define discrete principal contact element nets for the Plucker quadric and prove several elementary results. Employing a second real structure, we show that these results generalize previous results by Bobenko and Suris (2007) [18] on discrete Differential Geometry in the Lie quadric.
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On Discrete Differential Geometry in Twistor Space
arXiv: Differential Geometry, 2011Co-Authors: George ShapiroAbstract:In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in $S^4$ to complex values of a generalized cross-ratio by considering $S^4$ as a real section of the complex Pl\"ucker quadric, realized as the space of two-spheres in $S^4.$ We develop the Geometry of the Pl\"ucker quadric by examining the novel contact properties of two-spheres in $S^4,$ generalizing classical Lie Geometry in $S^3.$ Discrete Differential Geometry aims to develop discrete equivalents of the geometric notions and methods of classical Differential Geometry. We define discrete principal contact element nets for the Pl\"ucker quadric and prove several elementary results. Employing a second real real structure, we show that these results generalize previous results by Bobenko and Suris $(2007)$ on discrete Differential Geometry in the Lie quadric.
