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Bilinear Filtering
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 the other 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] |
Parallel (geometry)
In geometry, parallel lines are coplanar straight lines that do not intersect at any point. Parallel planes are planes in the same three-dimensional space that never meet. ''Parallel curves'' are curves that do not touch each other or intersect and keep a fixed minimum distance. In three-dimensional Euclidean space, a line and a plane that do not share a point are also said to be parallel. However, two noncoplanar lines are called ''skew lines''. Parallel lines are the subject of Euclid's parallel postulate. Parallelism is primarily a property of affine geometries and Euclidean geometry is a special instance of this type of geometry. In some other geometries, such as hyperbolic geometry, lines can have analogous properties that are referred to as parallelism. Symbol The parallel symbol is \parallel. For example, AB \parallel CD indicates that line ''AB'' is parallel to line ''CD''. In the Unicode character set, the "parallel" and "not parallel" signs have codepoint ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Texture Mapping
Texture mapping is a method for mapping a texture on a computer-generated graphic. Texture here can be high frequency detail, surface texture, or color. History The original technique was pioneered by Edwin Catmull in 1974. Texture mapping originally referred to diffuse mapping, a method that simply mapped pixels from a texture to a 3D surface ("wrapping" the image around the object). In recent decades, the advent of multi-pass rendering, multitexturing, mipmaps, and more complex mappings such as height mapping, bump mapping, normal mapping, displacement mapping, reflection mapping, specular mapping, occlusion mapping, and many other variations on the technique (controlled by a materials system) have made it possible to simulate near-photorealism in real time by vastly reducing the number of polygons and lighting calculations needed to construct a realistic and functional 3D scene. Texture maps A is an image applied (mapped) to the surface of a shape or polygon. This ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quadratic Formula
In elementary algebra, the quadratic formula is a formula that provides the solution(s) to a quadratic equation. There are other ways of solving a quadratic equation instead of using the quadratic formula, such as factoring (direct factoring, grouping, AC method), completing the square, graphing and others. Given a general quadratic equation of the form :ax^2+bx+c=0 with representing an unknown, with , and representing constants, and with , the quadratic formula is: :x = \frac where the plus–minus symbol "±" indicates that the quadratic equation has two solutions. Written separately, they become: : x_1=\frac\quad\text\quad x_2=\frac Each of these two solutions is also called a root (or zero) of the quadratic equation. Geometrically, these roots represent the -values at which ''any'' parabola, explicitly given as , crosses the -axis. As well as being a formula that yields the zeros of any parabola, the quadratic formula can also be used to identify the axis of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Grassman Product
In mathematics, the exterior algebra, or Grassmann algebra, named after Hermann Grassmann, is an algebra that uses the exterior product or wedge product as its multiplication. In mathematics, the exterior product or wedge product of vectors is an algebraic construction used in geometry to study areas, volumes, and their higher-dimensional analogues. The exterior product of two vectors u and v, denoted by u \wedge v, is called a bivector and lives in a space called the ''exterior square'', a vector space that is distinct from the original space of vectors. The magnitude of u \wedge v can be interpreted as the area of the parallelogram with sides u and v, which in three dimensions can also be computed using the cross product of the two vectors. More generally, all parallel plane surfaces with the same orientation and area have the same bivector as a measure of their oriented area. Like the cross product, the exterior product is anticommutative, meaning t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelogram
In Euclidean geometry, a parallelogram is a simple (non- self-intersecting) quadrilateral with two pairs of parallel sides. The opposite or facing sides of a parallelogram are of equal length and the opposite angles of a parallelogram are of equal measure. The congruence of opposite sides and opposite angles is a direct consequence of the Euclidean parallel postulate and neither condition can be proven without appealing to the Euclidean parallel postulate or one of its equivalent formulations. By comparison, a quadrilateral with just one pair of parallel sides is a trapezoid in American English or a trapezium in British English. The three-dimensional counterpart of a parallelogram is a parallelepiped. The etymology (in Greek παραλληλ-όγραμμον, ''parallēl-ógrammon'', a shape "of parallel lines") reflects the definition. Special cases *Rectangle – A parallelogram with four angles of equal size (right angles). *Rhombus – A parallelogram with four sides of eq ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Map
In projective geometry, a homography is an isomorphism of projective spaces, induced by an isomorphism of the vector spaces from which the projective spaces derive. It is a bijection that maps lines to lines, and thus a collineation. In general, some collineations are not homographies, but the fundamental theorem of projective geometry asserts that is not so in the case of real projective spaces of dimension at least two. Synonyms include projectivity, projective transformation, and projective collineation. Historically, homographies (and projective spaces) have been introduced to study perspective and projections in Euclidean geometry, and the term ''homography'', which, etymologically, roughly means "similar drawing", dates from this time. At the end of the 19th century, formal definitions of projective spaces were introduced, which differed from extending Euclidean or affine spaces by adding points at infinity. The term "projective transformation" originated in these abst ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hyperbola
In mathematics, a hyperbola (; pl. hyperbolas or hyperbolae ; adj. hyperbolic ) is a type of smooth curve lying in a plane, defined by its geometric properties or by equations for which it is the solution set. A hyperbola has two pieces, called connected components or branches, that are mirror images of each other and resemble two infinite bows. The hyperbola is one of the three kinds of conic section, formed by the intersection of a plane and a double cone. (The other conic sections are the parabola and the ellipse. A circle is a special case of an ellipse.) If the plane intersects both halves of the double cone but does not pass through the apex of the cones, then the conic is a hyperbola. Hyperbolas arise in many ways: * as the curve representing the reciprocal function y(x) = 1/x in the Cartesian plane, * as the path followed by the shadow of the tip of a sundial, * as the shape of an open orbit (as distinct from a closed elliptical orbit), such as the orbit of a s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convex Hull
In geometry, the convex hull or convex envelope or convex closure of a shape is the smallest convex set that contains it. The convex hull may be defined either as the intersection of all convex sets containing a given subset of a Euclidean space, or equivalently as the set of all convex combinations of points in the subset. For a bounded subset of the plane, the convex hull may be visualized as the shape enclosed by a rubber band stretched around the subset. Convex hulls of open sets are open, and convex hulls of compact sets are compact. Every compact convex set is the convex hull of its extreme points. The convex hull operator is an example of a closure operator, and every antimatroid can be represented by applying this closure operator to finite sets of points. The algorithmic problems of finding the convex hull of a finite set of points in the plane or other low-dimensional Euclidean spaces, and its dual problem of intersecting half-spaces, are fundamental problems of com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bézier Surface
Bézier surfaces are a species of mathematical spline used in computer graphics, computer-aided design, and finite element modeling. As with Bézier curves, a Bézier surface is defined by a set of control points. Similar to interpolation in many respects, a key difference is that the surface does not, in general, pass through the central control points; rather, it is "stretched" toward them as though each were an attractive force. They are visually intuitive, and for many applications, mathematically convenient. History Bézier surfaces were first described in 1962 by the French engineer Pierre Bézier who used them to design automobile bodies. Bézier surfaces can be of any degree, but bicubic Bézier surfaces generally provide enough degrees of freedom for most applications. Equation A given Bézier surface of degree (''n'', ''m'') is defined by a set of (''n'' + 1)(''m'' + 1) control points k''i'',''j'' where ''i'' = 0, ..., ''n'' and ''j'' = ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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