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Discrete Differential Geometry
Discrete differential geometry is the study of discrete counterparts of notions in differential geometry. Instead of smooth curves and surfaces, there are polygons, meshes, and simplicial complexes. It is used in the study of computer graphics Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. ..., geometry processing and topological combinatorics. See also * Discrete Laplace operator * Discrete exterior calculus * Discrete Morse theory * Topological combinatorics * Spectral shape analysis * Analysis on fractals * Discrete calculus ReferencesDiscrete differential geometry Forum* * * Alexander I. Bobenko, Yuri B. Suris (2008), "Discrete Differential Geometry", American Mathematical Society, Differential geometry Simplicial sets {{differential-geometry-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Differential Geometry
Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of Calculus, single variable calculus, vector calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as classical antiquity, antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Nikolai Lobachevsky, Lobachevsky. The simplest examples of smooth spaces are the Differential geometry of curves, plane and space curves and Differential geometry of surfaces, surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Polygon
In geometry, a polygon () is a plane figure made up of line segments connected to form a closed polygonal chain. The segments of a closed polygonal chain are called its '' edges'' or ''sides''. The points where two edges meet are the polygon's '' vertices'' or ''corners''. An ''n''-gon is a polygon with ''n'' sides; for example, a triangle is a 3-gon. A simple polygon is one which does not intersect itself. More precisely, the only allowed intersections among the line segments that make up the polygon are the shared endpoints of consecutive segments in the polygonal chain. A simple polygon is the boundary of a region of the plane that is called a ''solid polygon''. The interior of a solid polygon is its ''body'', also known as a ''polygonal region'' or ''polygonal area''. In contexts where one is concerned only with simple and solid polygons, a ''polygon'' may refer only to a simple polygon or to a solid polygon. A polygonal chain may cross over itself, creating star polyg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Polygon Mesh
In 3D computer graphics and solid modeling, a polygon mesh is a collection of , s and s that defines the shape of a polyhedron, polyhedral object's surface. It simplifies Rendering (computer graphics), rendering, as in a wire-frame model. The face (geometry), faces usually consist of triangles (triangle mesh), quadrilaterals (quads), or other simple convex polygon, convex polygons (n-gons). A polygonal mesh may also be more generally composed of concave polygon, concave polygons, or even Polygon with holes, polygons with holes. The study of Polygon (computer graphics), polygon meshes is a large sub-field of computer graphics (specifically 3D computer graphics) and geometric modeling. Different representations of polygon meshes are used for different applications and goals. The variety of operations performed on meshes includes Boolean logic (Constructive solid geometry), Subdivision surfaces, smoothing, and Level of detail (computer graphics), simplification. Algorithms also exist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Simplicial Complexes
In mathematics, a simplicial complex is a structured set composed of points, line segments, triangles, and their ''n''-dimensional counterparts, called simplices, such that all the faces and intersections of the elements are also included in the set (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex. To distinguish a simplicial complex from an abstract simplicial complex, the former is often called a geometric simplicial complex., Section 4.3 Definitions A simplicial complex \mathcal is a set of simplices that satisfies the following conditions: # Every face of a simplex from \mathcal is also in \mathcal. # The non-empty intersection of any two simplices \sigma_1, \sigma_2 \in \mathcal is a face of both \sigma_1 and \sigma_2. See also the definition of an abstract ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Computer Graphics
Computer graphics deals with generating images and art with the aid of computers. Computer graphics is a core technology in digital photography, film, video games, digital art, cell phone and computer displays, and many specialized applications. A great deal of specialized hardware and software has been developed, with the displays of most devices being driven by graphics hardware, computer graphics hardware. It is a vast and recently developed area of computer science. The phrase was coined in 1960 by computer graphics researchers Verne Hudson and William Fetter of Boeing. It is often abbreviated as CG, or typically in the context of film as Computer-generated imagery, computer generated imagery (CGI). The non-artistic aspects of computer graphics are the subject of Computer graphics (computer science), computer science research. Some topics in computer graphics include user interface design, Sprite (computer graphics), sprite graphics, raster graphics, Rendering (computer graph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Geometry Processing
Geometry processing is an area of research that uses concepts from applied mathematics, computer science and engineering to design efficient algorithms for the acquisition, 3D reconstruction, reconstruction, analysis, manipulation, simulation and transmission of complex 3D models. As the name implies, many of the concepts, data structures, and algorithms are directly analogous to signal processing and image processing. For example, where image smoothing might convolve an intensity signal with a blur kernel formed using the Laplace operator, Laplacian smoothing, geometric smoothing might be achieved by convolving a Surface (mathematics), surface geometry with a blur kernel formed using the Laplace-Beltrami operator. Applications of geometry processing algorithms already cover a wide range of areas from multimedia, entertainment and classical computer-aided design, to biomedical computing, reverse engineering, and scientific computing. Geometry processing is a common research topi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Topological Combinatorics
The mathematical discipline of topological combinatorics is the application of topological and algebro-topological methods to solving problems in combinatorics. History The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this turned into the field of algebraic topology. In 1978 the situation was reversed—methods from algebraic topology were used to solve a problem in combinatorics—when László Lovász proved the Kneser conjecture, thus beginning the new field of topological combinatorics. Lovász's proof used the Borsuk–Ulam theorem and this theorem retains a prominent role in this new field. This theorem has many equivalent versions and analogs and has been used in the study of fair division problems. In another application of homological methods to graph theory, Lovász proved both the undirected and directed versions of a conjecture of András Frank: Given a ''k''-connected graph ''G'', ''k'' points v_1,\l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Discrete Laplace Operator
In mathematics, the discrete Laplace operator is an analog of the continuous Laplace operator, defined so that it has meaning on a Graph (discrete mathematics), graph or a lattice (group), discrete grid. For the case of a finite-dimensional graph (having a finite number of edges and vertices), the discrete Laplace operator is more commonly called the Laplacian matrix. The discrete Laplace operator occurs in physics problems such as the Ising model and loop quantum gravity, as well as in the study of discrete dynamical systems. It is also used in numerical analysis as a stand-in for the continuous Laplace operator. Common applications include image processing, where it is known as the Laplace filter, and in machine learning for cluster analysis, clustering and semi-supervised learning on neighborhood graphs. Definitions Graph Laplacians There are various definitions of the ''discrete Laplacian'' for Graph (discrete mathematics), graphs, differing by sign and scale factor (sometim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Discrete Exterior Calculus
In mathematics, the discrete exterior calculus (DEC) is the extension of the exterior calculus to discrete spaces including graphs, finite element meshes, and lately also general polygonal meshes (non-flat and non-convex). DEC methods have proved to be very powerful in improving and analyzing finite element methods: for instance, DEC-based methods allow the use of highly non-uniform meshes to obtain accurate results. Non-uniform meshes are advantageous because they allow the use of large elements where the process to be simulated is relatively simple, as opposed to a fine resolution where the process may be complicated (e.g., near an obstruction to a fluid flow), while using less computational power than if a uniformly fine mesh were used. The discrete exterior derivative Stokes' theorem relates the integral of a differential (''n'' − 1)-form ''ω'' over the boundary ∂''M'' of an ''n''-dimensional manifold ''M'' to the integral of d''ω'' (the exterior derivativ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Discrete Morse Theory
Discrete Morse theory is a combinatorial adaptation of Morse theory developed by Robin Forman. The theory has various practical applications in diverse fields of applied mathematics and computer science, such as configuration spaces, homology computation, denoising, mesh compression, and topological data analysis. Notation regarding CW complexes Let X be a CW complex and denote by \mathcal its set of cells. Define the ''incidence function'' \kappa\colon\mathcal \times \mathcal \to \mathbb in the following way: given two cells \sigma and \tau in \mathcal, let \kappa(\sigma,~\tau) be the degree of the attaching map from the boundary of \sigma to \tau. The boundary operator is the endomorphism \partial of the free abelian group generated by \mathcal defined by :\partial(\sigma) = \sum_\kappa(\sigma,\tau)\tau. It is a defining property of boundary operators that \partial\circ\partial \equiv 0. In more axiomatic definitions one can find the requirement that \forall \sigma,\tau^ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Topological Combinatorics
The mathematical discipline of topological combinatorics is the application of topological and algebro-topological methods to solving problems in combinatorics. History The discipline of combinatorial topology used combinatorial concepts in topology and in the early 20th century this turned into the field of algebraic topology. In 1978 the situation was reversed—methods from algebraic topology were used to solve a problem in combinatorics—when László Lovász proved the Kneser conjecture, thus beginning the new field of topological combinatorics. Lovász's proof used the Borsuk–Ulam theorem and this theorem retains a prominent role in this new field. This theorem has many equivalent versions and analogs and has been used in the study of fair division problems. In another application of homological methods to graph theory, Lovász proved both the undirected and directed versions of a conjecture of András Frank: Given a ''k''-connected graph ''G'', ''k'' points v_1,\l ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
Spectral Shape Analysis
Spectral shape analysis relies on the spectrum (eigenvalues and/or eigenfunctions) of the Laplace–Beltrami operator to compare and analyze geometric shapes. Since the spectrum of the Laplace–Beltrami operator is invariant under isometries, it is well suited for the analysis or retrieval of non-rigid shapes, i.e. bendable objects such as humans, animals, plants, etc. Laplace The Laplace–Beltrami operator is involved in many important differential equations, such as the heat equation and the wave equation. It can be defined on a Riemannian manifold as the divergence of the gradient of a real-valued function ''f'': :\Delta f := \operatorname \operatorname f. Its spectral components can be computed by solving the Helmholtz equation (or Laplacian eigenvalue problem): : \Delta \varphi_i + \lambda_i \varphi_i = 0. The solutions are the eigenfunctions \varphi_i (modes) and corresponding eigenvalues \lambda_i, representing a diverging sequence of positive real numbers. The first ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] [Amazon] |
