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Triangulation (geometry)
In geometry, a triangulation is a subdivision of a planar object into triangles, and by extension the subdivision of a higher-dimension geometric object into simplices. Triangulations of a three-dimensional volume would involve subdividing it into tetrahedra packed together. In most instances, the triangles of a triangulation are required to meet edge-to-edge and vertex-to-vertex. Types Different types of triangulations may be defined, depending both on what geometric object is to be subdivided and on how the subdivision is determined. * A triangulation T of \mathbb^d is a subdivision of \mathbb^d into d-dimensional simplices such that any two simplices in T intersect in a common face (a simplex of any lower dimension) or not at all, and any bounded set in \mathbb^d intersects only finitely many simplices in T. That is, it is a locally finite simplicial complex that covers the entire space. * A point-set triangulation, i.e., a triangulation of a discrete set of points \mathcal\su ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometry
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by the algorithm, supposing that each elementary operation takes a fixed amount of time to perform. Thus, the amount of time taken and the number of elementary operations performed by the algorithm are taken to be related by a constant factor. Since an algorithm's running time may vary among different inputs of the same size, one commonly considers the worst-case time complexity, which is the maximum amount of time required for inputs of a given size. Less common, and usually specified explicitly, is the average-case complexity, which is the average of the time taken on inputs of a given size (this makes sense because there are only a finite number of possible inputs of a given size). In both cases, the time complexity is generally expresse ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ruppert's Algorithm
In mesh generation, Delaunay refinement are algorithms for mesh generation based on the principle of adding Steiner points to the geometry of an input to be meshed, in a way that causes the Delaunay triangulation or constrained Delaunay triangulation of the augmented input to meet the quality requirements of the meshing application. Delaunay refinement methods include methods by Chew and by Ruppert. Chew's second algorithm Chew's second algorithm takes a piecewise linear system (PLS) and returns a constrained Delaunay triangulation of only quality triangles where quality is defined by the minimum angle in a triangle. Developed by L. Paul Chew for meshing surfaces embedded in three-dimensional space, Chew's second algorithm has been adopted as a two-dimensional mesh generator due to practical advantages over Ruppert's algorithm in certain cases and is the default quality mesh generator implemented in the freely available Triangle package. Chew's second algorithm is guaranteed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chew's Second Algorithm
In mesh generation, Delaunay refinement are algorithms for mesh generation based on the principle of adding Steiner points to the geometry of an input to be meshed, in a way that causes the Delaunay triangulation or constrained Delaunay triangulation of the augmented input to meet the quality requirements of the meshing application. Delaunay refinement methods include methods by Chew and by Ruppert. Chew's second algorithm Chew's second algorithm takes a piecewise linear system (PLS) and returns a constrained Delaunay triangulation of only quality triangles where quality is defined by the minimum angle in a triangle. Developed by L. Paul Chew for meshing surfaces embedded in three-dimensional space, Chew's second algorithm has been adopted as a two-dimensional mesh generator due to practical advantages over Ruppert's algorithm in certain cases and is the default quality mesh generator implemented in the freely available Triangle package. Chew's second algorithm is guaranteed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Delaunay Refinement
In mesh generation, Delaunay refinement are algorithms for mesh generation based on the principle of adding Steiner points to the geometry of an input to be meshed, in a way that causes the Delaunay triangulation or constrained Delaunay triangulation of the augmented input to meet the quality requirements of the meshing application. Delaunay refinement methods include methods by Chew and by Ruppert. Chew's second algorithm Chew's second algorithm takes a piecewise linear system (PLS) and returns a constrained Delaunay triangulation of only quality triangles where quality is defined by the minimum angle in a triangle. Developed by L. Paul Chew for meshing surfaces embedded in three-dimensional space, Chew's second algorithm has been adopted as a two-dimensional mesh generator due to practical advantages over Ruppert's algorithm in certain cases and is the default quality mesh generator implemented in the freely available Triangle package. Chew's second algorithm is guaranteed to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nonobtuse Mesh
In computer graphics, a nonobtuse triangle mesh is a polygon mesh composed of a set of triangles in which no angle is obtuse, ''i.e.'' greater than 90°. If each (triangle) face angle is strictly less than 90°, then the triangle mesh is said to be acute. Every polygon with n sides has a nonobtuse triangulation with O(n) triangles (expressed in big O notation), allowing some triangle vertices to be added to the sides and interior of the polygon. These nonobtuse triangulations can be further refined to produce acute triangulations with O(n) triangles. Nonobtuse meshes avoid certain problems of nonconvergence or of convergence to the wrong numerical solution as demonstrated by the Schwarz lantern. The immediate benefits of a nonobtuse or acute mesh include more efficient and more accurate geodesic computation using fast marching, and guaranteed validity for planar mesh embeddings via discrete harmonic maps. References {{reflist See also *Triangle mesh In computer graphics, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Types Of Mesh
A mesh is a representation of a larger geometric domain by smaller discrete cells. Meshes are commonly used to compute solutions of partial differential equations and render computer graphics, and to analyze geographical and cartographic data. A mesh partitions space into ''elements'' (or ''cells'' or ''zones'') over which the equations can be solved, which then approximates the solution over the larger domain. Element boundaries may be constrained to lie on internal or external boundaries within a model. Higher-quality (better-shaped) elements have better numerical properties, where what constitutes a "better" element depends on the general governing equations and the particular solution to the model instance. Common cell shapes Two-dimensional There are two types of two-dimensional cell shapes that are commonly used. These are the triangle and the quadrilateral. Computationally poor elements will have sharp internal angles or short edges or both. Triangle This cell shape consis ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Steiner Point (computational Geometry)
In computational geometry, a Steiner point is a point that is not part of the input to a geometric optimization problem but is added during the solution of the problem, to create a better solution than would be possible from the original points alone. The name of these points comes from the Steiner tree problem, named after Jakob Steiner, in which the goal is to connect the input points by a network of minimum total length. If the input points alone are used as endpoints of the network edges, then the shortest network is their minimum spanning tree. However, shorter networks can often be obtained by adding Steiner points, and using both the new points and the input points as edge endpoints. Another problem that uses Steiner points is Steiner triangulation. The goal is to partition an input (such as a point set or polygon) into triangles, meeting edge-to-edge. Both input points and Steiner points may be used as triangle vertices. See also *Delaunay refinement In mesh generation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangle Mesh
In computer graphics, a triangle mesh is a type of polygon mesh. It comprises a set of triangles (typically in three dimensions) that are connected by their common edges or vertices. Many graphics software packages and hardware devices can operate more efficiently on triangles that are grouped into meshes than on a similar number of triangles that are presented individually. This is typically because computer graphics do operations on the vertices at the corners of triangles. With individual triangles, the system has to operate on three vertices for every triangle. In a large mesh, there could be eight or more triangles meeting at a single vertex - by processing those vertices just once, it is possible to do a fraction of the work and achieve an identical effect. In many computer graphics applications it is necessary to manage a mesh of triangles. The mesh components are vertices, edges, and triangles. An application might require knowledge of the various connections bet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 polyhedral object. The faces usually consist of triangles (triangle mesh), quadrilaterals (quads), or other simple convex polygons ( n-gons), since this simplifies rendering, but may also be more generally composed of concave polygons, or even polygons with holes. The study of 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 may include: Boolean logic ( Constructive solid geometry), smoothing, simplification, and many others. Algorithms also exist for ray tracing, collision detection, and rigid-body dynamics with polygon meshes. If the mesh's edges are rendered instead of the faces, then the model becomes a wireframe model. Volumetric meshes are distinct f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Element Method
The finite element method (FEM) is a popular method for numerically solving differential equations arising in engineering and mathematical modeling. Typical problem areas of interest include the traditional fields of structural analysis, heat transfer, fluid flow, mass transport, and electromagnetic potential. The FEM is a general numerical method for solving partial differential equations in two or three space variables (i.e., some boundary value problems). To solve a problem, the FEM subdivides a large system into smaller, simpler parts that are called finite elements. This is achieved by a particular space discretization in the space dimensions, which is implemented by the construction of a mesh of the object: the numerical domain for the solution, which has a finite number of points. The finite element method formulation of a boundary value problem finally results in a system of algebraic equations. The method approximates the unknown function over the domain. The sim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Surface Triangulation
Triangulation of a surface means * a ''net'' of triangles, which covers a given surface partly or totally, ''or'' * the ''procedure'' of generating the points and triangles of such a net of triangles. Approaches This article describes the generation of a net of triangles. In literature there are contributions which deal with the optimization of a given net. Surface triangulations are important for * visualizing surfaces and * the application of finite element methods. The triangulation of a ''parametrically'' defined surface is simply achieved by triangulating the area of definition (see second figure, depicting the Monkey Saddle). However, the triangles may vary in shape and extension in object space, posing a potential drawback. This can be minimized through adaptive methods that consider step width while triangulating the parameter area. To triangulate an ''implicit surface'' (defined by one or more equations) is more difficult. There exist essentially two methods. * On ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
