Trilinear Poles Of A Pencil Of Lines
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 * Trilinear polarity In Euclidean geometry, trilinear polarity is a certain correspondence between the points in the Plane (geometry), plane of a triangle not lying on the sides of the triangle and lines in the plane of the triangle not passing through the Vertex (g ...

Trilinear Filtering
Trilinear filtering is an extension of the bilinear texture filtering method, which also performs linear interpolation between mipmaps. Bilinear filtering has several weaknesses that make it an unattractive choice in many cases: using it on a full-detail texture when scaling to a very small size causes accuracy problems from missed texels, and compensating for this by using multiple mipmaps throughout the polygon leads to abrupt changes in blurriness, which is most pronounced in polygons that are steeply angled relative to the camera. To solve this problem, trilinear filtering interpolates between the results of bilinear filtering on the two mipmaps nearest to the detail required for the polygon at the pixel. If the pixel would take up 1/100 of the texture in one direction, trilinear filtering would interpolate between the result of filtering the 128×128 mipmap as y1 with x1 as 128, and the result of filtering on the 64×64 mipmap as y2 with x2 as 64, and then interpolate to { ...
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Trilinear Form
In abstract algebra and multilinear algebra, a multilinear form on a vector space V over a field K is a map :f\colon V^k \to K that is separately ''K''-linear in each of its ''k'' arguments. More generally, one can define multilinear forms on a module over a commutative ring. The rest of this article, however, will only consider multilinear forms on finite-dimensional vector spaces. A multilinear ''k''-form on V over \mathbf is called a (covariant) ''k''-tensor, and the vector space of such forms is usually denoted \mathcal^k(V) or \mathcal^k(V). Tensor product Given a ''k''-tensor f\in\mathcal^k(V) and an ''ℓ''-tensor g\in\mathcal^\ell(V), a product f\otimes g\in\mathcal^(V), known as the tensor product, can be defined by the property : (f\otimes g)(v_1,\ldots,v_k,v_,\ldots, v_)=f(v_1,\ldots,v_k)g(v_,\ldots, v_), for all v_1,\ldots,v_\in V. The tensor product of multilinear forms is not commutative; however it is bilinear and associative: : f\otimes(ag_1+bg_2)=a(f\otime ...
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Trilinear Interpolation
Trilinear interpolation is a method of multivariate interpolation on a 3-dimensional regular grid. It approximates the value of a function at an intermediate point (x, y, z) within the local axial rectangular prism linearly, using function data on the lattice points. For an arbitrary, unstructured mesh (as used in finite element analysis), other methods of interpolation must be used; if all the mesh elements are tetrahedra (3D simplices), then barycentric coordinates provide a straightforward procedure. Trilinear interpolation is frequently used in numerical analysis, data analysis, and computer graphics. Compared to linear and bilinear interpolation 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-defi ...
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Trilinear Map
In linear algebra, a multilinear map is a function of several variables that is linear separately in each variable. More precisely, a multilinear map is a function :f\colon V_1 \times \cdots \times V_n \to W\text where V_1,\ldots,V_n and W are vector spaces (or modules over a commutative ring), with the following property: for each i, if all of the variables but v_i are held constant, then f(v_1, \ldots, v_i, \ldots, v_n) is a linear function of v_i. A multilinear map of one variable is a linear map, and of two variables is a bilinear map. More generally, a multilinear map of ''k'' variables is called a ''k''-linear map. If the codomain of a multilinear map is the field of scalars, it is called a multilinear form. Multilinear maps and multilinear forms are fundamental objects of study in multilinear algebra. If all variables belong to the same space, one can consider symmetric, antisymmetric and alternating ''k''-linear maps. The latter coincide if the underlying ring ...
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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 the ratio of the perpendicular distances from the point to the sides (extended if necessary) opposite vertices and respectively; the ratio is the ratio of the perpendicular distances from the point to the sidelines opposite vertices and respectively; and likewise for and vertices and . In the diagram at right, the trilinear coordinates of the indicated interior point are the actual distances (, , ), or equivalently in ratio form, for any positive constant . If a point is on a sideline of the reference triangle, its corresponding trilinear coordinate is 0. If an exterior point is on the opposite side of a sideline from the interior of the triangle, its trilinear coordinate associated with that sideline is negative. It is impossible ...
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