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Trilinear Interpolation
Trilinear interpolation is a method of multivariate interpolation on a Three dimensional space, 3-dimensional regular grid. It approximates the value of a function at an intermediate point (x, y, z) within the local axial rectangular prism (geometry), prism linearly, using function data on the lattice points. Trilinear interpolation is frequently used in numerical analysis, data analysis, and computer graphics. Related methods Trilinear interpolation is the extension of linear interpolation, which operates in spaces with dimension D = 1, and bilinear interpolation, which operates with dimension D = 2, to dimension D = 3. These interpolation schemes all use polynomials of order 1, giving an accuracy of order 2, and it requires 2^D = 8 adjacent pre-defined values surrounding the interpolation point. There are several ways to arrive at trilinear interpolation, which is equivalent to 3-dimensional tensor B-spline interpolation of order 1, and the trilinear interpolation operator ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
3D Interpolation2
3D, 3-D, 3d, or Three D may refer to: Science, technology, and mathematics * A three-dimensional space in mathematics Relating to three-dimensionality * 3D computer graphics, computer graphics that use a three-dimensional representation of geometric data * 3D display, a type of information display that conveys depth to the viewer * 3D film, a motion picture that gives the illusion of three-dimensional perception * 3D modeling, developing a representation of any three-dimensional surface or object * 3D printing, making a three-dimensional solid object of a shape from a digital model * 3D television, television that conveys depth perception to the viewer * 3D projection * 3D rendering * 3D scanning, making a digital representation of three-dimensional objects * Video game graphics#3D, 3D video game * Stereoscopy, any technique capable of recording three-dimensional visual information or creating the illusion of depth in an image * Three-dimensional space Other uses in science and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Element
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. Computers are usually used to perform the calculations required. With high-speed supercomputers, better solutions can be achieved and are often required to solve the largest and most complex problems. FEM is a general numerical method for solving partial differential equations in two- or three-space variables (i.e., some boundary value problems). There are also studies about using FEM to solve high-dimensional problems. To solve a problem, FEM subdivides a large system into smaller, simpler parts 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 numerica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Slerp
In computer graphics, slerp is shorthand for spherical linear interpolation, introduced by Ken Shoemake in the context of quaternion interpolation for the purpose of animating 3D rotation. It refers to constant-speed motion along a unit-radius great circle arc, given the ends and an interpolation parameter between 0 and 1. Geometric slerp Slerp has a geometric formula independent of quaternions, and independent of the dimension of the space in which the arc is embedded. This formula, a symmetric weighted sum credited to Glenn Davis, is based on the fact that any point on the curve must be a linear combination of the ends. Let ''p''0 and ''p''1 be the first and last points of the arc, and let ''t'' be the parameter, . Compute Ω as the angle subtended by the arc, so that , the ''n''-dimensional dot product of the unit vectors from the origin to the ends. The geometric formula is then : \operatorname(p_0,p_1; t) = \frac p_0 + \frac p_1. The symmetry lies in the fact that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedral Interpolation
In geometry, a tetrahedron (: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertices. The tetrahedron is the simplest of all the ordinary convex polyhedra. The tetrahedron is the three-dimensional case of the more general concept of a Euclidean simplex, and may thus also be called a 3-simplex. The tetrahedron is one kind of pyramid, which is a polyhedron with a flat polygon base and triangular faces connecting the base to a common point. In the case of a tetrahedron, the base is a triangle (any of the four faces can be considered the base), so a tetrahedron is also known as a "triangular pyramid". Like all convex polyhedra, a tetrahedron can be folded from a single sheet of paper. It has two such nets. For any tetrahedron there exists a sphere (called the circumsphere) on which all four vertices lie, and another sphere (the insphere) tangent to the tetrahedron's faces. Re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radial Interpolation
Radial is a geometric term of location which may refer to: Mathematics and Direction * Vector (geometric), a line * Radius, adjective form of * Radial distance (geometry), a directional coordinate in a polar coordinate system * Radial set * A bearing from a waypoint, such as a VHF omnidirectional range Very High Frequency Omnidirectional Range Station (VOR) is a type of short-range VHF radio navigation system for aircraft, enabling aircraft with a VOR receiver to determine the azimuth (also radial), referenced to magnetic north, between the a ... Biology * Radial artery, the main artery of the lateral aspect of the forearm * Radial nerve, supplies the posterior portion of the upper limb * Radial symmetry, one of the types of distribution of body parts or shapes in biology * Radius (bone), a bone of the forearm Technology * Radial (radio), lines which radiate from a radio antenna * Radial axle, on a locomotive or carriage * Radial compressor * Radial delayed blowback * Ra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tricubic Interpolation
In the mathematical subfield numerical analysis, tricubic interpolation is a method for obtaining values at arbitrary points in Three-dimensional space, 3D space of a function defined on a regular grid. The approach involves approximating the function locally by an expression of the form f(x,y,z)=\sum_^3 \sum_^3 \sum_^3 a_ x^i y^j z^k. This form has 64 coefficients a_; requiring the function to have a given value or given directional derivative at a point places one linear constraint on the 64 coefficients. The term ''tricubic interpolation'' is used in more than one context; some experiments measure both the value of a function and its spatial derivatives, and it is desirable to interpolate preserving the values and the measured derivatives at the grid points. Those provide 32 constraints on the coefficients, and another 32 constraints can be provided by requiring smoothness of higher derivatives. In other contexts, we can obtain the 64 coefficients by considering a 3×3 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilinear Interpolation
In mathematics, bilinear interpolation is a method for interpolating functions of two variables (e.g., ''x'' and ''y'') using repeated linear interpolation. It is usually applied to functions sampled on a 2D rectilinear grid, though it can be generalized to functions defined on the vertices of (a mesh of) arbitrary convex quadrilaterals. Bilinear interpolation is performed using linear interpolation first in one direction, and then again in another direction. Although each step is linear in the sampled values and in the position, the interpolation as a whole is not linear but rather quadratic in the sample location. Bilinear interpolation is one of the basic resampling techniques in computer vision and image processing, where it is also called bilinear filtering or bilinear texture mapping. Computation Suppose that we want to find the value of the unknown function ''f'' at the point (''x'', ''y''). It is assumed that we know the value of ''f'' at the four points ''Q''11 = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Interpolation
In mathematics, linear interpolation is a method of curve fitting using linear polynomials to construct new data points within the range of a discrete set of known data points. Linear interpolation between two known points If the two known points are given by the coordinates (x_0,y_0) and the linear interpolant is the straight line between these points. For a value x in the interval the value y along the straight line is given from the equation of slopes \frac = \frac, which can be derived geometrically from the figure on the right. It is a special case of polynomial interpolation with Solving this equation for y, which is the unknown value at x, gives \begin y &= y_0 + (x-x_0)\frac \\ &= \frac + \frac\\ &= \frac \\ &= \frac, \end which is the formula for linear interpolation in the interval Outside this interval, the formula is identical to linear extrapolation. This formula can also be understood as a weighted average. The weights are inversely related to the dist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trilinear Interpolation Visualisation
Trilinear may refer to: * Trilinear filtering, a method in computer graphics for choosing the color of a texture * Trilinear form, a type of mathematical function from a vector space to the underlying field * Trilinear interpolation, an extension of linear interpolation for interpolating functions of three variables on a rectilinear 3D grid * Trilinear map, a type of mathematical function between vector spaces * Trilinear coordinates In geometry, the trilinear coordinates of a point relative to a given triangle describe the relative directed distances from the three sidelines of the triangle. Trilinear coordinates are an example of homogeneous coordinates. The ratio is ... * Trilinear polarity, in geometry {{disambiguation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Enclosing Points
Enclosure or inclosure is a term, used in English landownership, that refers to the appropriation of "waste" or "common land", enclosing it, and by doing so depriving commoners of their traditional rights of access and usage. Agreements to enclose land could be either through a formal or informal process. The process could normally be accomplished in three ways. First there was the creation of "closes", taken out of larger common fields by their owners. Secondly, there was enclosure by proprietors, owners who acted together, usually small farmers or squires, leading to the enclosure of whole parishes. Finally there were inclosure act, enclosures by acts of Parliament. The stated justification for enclosure was to improve the efficiency of agriculture. However, there were other motives too, one example being that the value of the land enclosed would be substantially increased. There were social consequences to the policy, with many protests at the removal of rights from the comm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Barycentric Coordinates (mathematics)
In geometry, a barycentric coordinate system is a coordinate system in which the location of a point is specified by reference to a simplex (a triangle for points in a plane, a tetrahedron for points in three-dimensional space, etc.). The barycentric coordinates of a point can be interpreted as masses placed at the vertices of the simplex, such that the point is the center of mass (or ''barycenter'') of these masses. These masses can be zero or negative; they are all positive if and only if the point is inside the simplex. Every point has barycentric coordinates, and their sum is never zero. Two tuples of barycentric coordinates specify the same point if and only if they are proportional; that is to say, if one tuple can be obtained by multiplying the elements of the other tuple by the same non-zero number. Therefore, barycentric coordinates are either considered to be defined up to multiplication by a nonzero constant, or normalized for summing to unity. Barycentric coordin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Simplex
In geometry, a simplex (plural: simplexes or simplices) is a generalization of the notion of a triangle or tetrahedron to arbitrary dimensions. The simplex is so-named because it represents the simplest possible polytope in any given dimension. For example, * a 0-dimensional simplex is a point, * a 1-dimensional simplex is a line segment, * a 2-dimensional simplex is a triangle, * a 3-dimensional simplex is a tetrahedron, and * a 4-dimensional simplex is a 5-cell. Specifically, a -simplex is a -dimensional polytope that is the convex hull of its vertices. More formally, suppose the points u_0, \dots, u_k are affinely independent, which means that the vectors u_1 - u_0,\dots, u_k-u_0 are linearly independent. Then, the simplex determined by them is the set of points C = \left\. A regular simplex is a simplex that is also a regular polytope. A regular -simplex may be constructed from a regular -simplex by connecting a new vertex to all original vertices by the common ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
