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Line–line Intersection
In Euclidean geometry, the intersection of a line and a line can be the empty set, a point (geometry), point, or another Line (geometry), line. Distinguishing these cases and finding the Intersection (Euclidean geometry), intersection have uses, for example, in computer graphics, motion planning, and collision detection. In three-dimensional Euclidean geometry, if two lines are not in the same plane (geometry), plane, they have no point of intersection and are called skew lines. If they are in the same plane, however, there are three possibilities: if they coincide (are not distinct lines), they have an infinitude of points in common (namely all of the points on either of them); if they are distinct but have the same slope, they are said to be parallel (geometry), parallel and have no points in common; otherwise, they have a single point of intersection. The distinguishing features of non-Euclidean geometry are the number and locations of possible intersections between two lines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Intersection (Euclidean Geometry)
In geometry, an intersection is a point, line, or curve common to two or more objects (such as lines, curves, planes, and surfaces). The simplest case in Euclidean geometry is the line–line intersection between two distinct line (geometry), lines, which either is one point (geometry), point or does not exist (if the lines are parallel lines, parallel). Other types of geometric intersection include: * Line–plane intersection * Line–sphere intersection * Intersection of a polyhedron with a line * Line segment intersection * Intersection curve Determination of the intersection of flat (geometry), flats – linear geometric objects embedded in a higher-dimensional space – is a simple task of linear algebra, namely the solution of a system of linear equations. In general the determination of an intersection leads to non-linear equations, which can be numerical solution, solved numerically, for example using Newton iteration. Intersection problems between a line and a conic sec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tetrahedron
In geometry, a tetrahedron (plural: tetrahedra or tetrahedrons), also known as a triangular pyramid, is a polyhedron composed of four triangular faces, six straight edges, and four vertex corners. The tetrahedron is the simplest of all the ordinary convex polyhedra and the only one that has fewer than 5 faces. 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Unit Vector
In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1. A unit vector is often denoted by a lowercase letter with a circumflex, or "hat", as in \hat (pronounced "v-hat"). The term ''direction vector'', commonly denoted as d, is used to describe a unit vector being used to represent spatial direction and relative direction. 2D spatial directions are numerically equivalent to points on the unit circle and spatial directions in 3D are equivalent to a point on the unit sphere. The normalized vector û of a non-zero vector u is the unit vector in the direction of u, i.e., :\mathbf = \frac where , u, is the norm (or length) of u. The term ''normalized vector'' is sometimes used as a synonym for ''unit vector''. Unit vectors are often chosen to form the basis of a vector space, and every vector in the space may be written as a linear combination of unit vectors. Orthogonal coordinates Cartesian coordinates Unit vectors may be us ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Least-squares
The method of least squares is a standard approach in regression analysis to approximate the solution of overdetermined systems (sets of equations in which there are more equations than unknowns) by minimizing the sum of the squares of the residuals (a residual being the difference between an observed value and the fitted value provided by a model) made in the results of each individual equation. The most important application is in data fitting. When the problem has substantial uncertainties in the independent variable (the ''x'' variable), then simple regression and least-squares methods have problems; in such cases, the methodology required for fitting errors-in-variables models may be considered instead of that for least squares. Least squares problems fall into two categories: linear or ordinary least squares and nonlinear least squares, depending on whether or not the residuals are linear in all unknowns. The linear least-squares problem occurs in statistical regression ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Moore–Penrose Inverse
In mathematics, and in particular linear algebra, the Moore–Penrose inverse of a matrix is the most widely known generalization of the inverse matrix. It was independently described by E. H. Moore in 1920, Arne Bjerhammar in 1951, and Roger Penrose in 1955. Earlier, Erik Ivar Fredholm had introduced the concept of a pseudoinverse of integral operators in 1903. When referring to a matrix, the term pseudoinverse, without further specification, is often used to indicate the Moore–Penrose inverse. The term generalized inverse is sometimes used as a synonym for pseudoinverse. A common use of the pseudoinverse is to compute a "best fit" ( least squares) solution to a system of linear equations that lacks a solution (see below under § Applications). Another use is to find the minimum ( Euclidean) norm solution to a system of linear equations with multiple solutions. The pseudoinverse facilitates the statement and proof of results in linear algebra. The pseudoinverse is defined a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Augmented Matrix
In linear algebra, an augmented matrix is a matrix obtained by appending the columns of two given matrices, usually for the purpose of performing the same elementary row operations on each of the given matrices. Given the matrices and , where A = \begin 1 & 3 & 2 \\ 2 & 0 & 1 \\ 5 & 2 & 2 \end , \quad B = \begin 4 \\ 3 \\ 1 \end, the augmented matrix (''A'', ''B'') is written as (A, B) = \left begin 1 & 3 & 2 & 4 \\ 2 & 0 & 1 & 3 \\ 5 & 2 & 2 & 1 \end\right This is useful when solving systems of linear equations. For a given number of unknowns, the number of solutions to a system of linear equations depends only on the rank of the matrix representing the system and the rank of the corresponding augmented matrix. Specifically, according to the Rouché–Capelli theorem, any system of linear equations is inconsistent (has no solutions) if the rank of the augmented matrix is greater than the rank of the coefficient matrix; if, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Rank
In linear algebra, the rank of a matrix is the dimension of the vector space generated (or spanned) by its columns. p. 48, § 1.16 This corresponds to the maximal number of linearly independent columns of . This, in turn, is identical to the dimension of the vector space spanned by its rows. Rank is thus a measure of the " nondegenerateness" of the system of linear equations and linear transformation encoded by . There are multiple equivalent definitions of rank. A matrix's rank is one of its most fundamental characteristics. The rank is commonly denoted by or ; sometimes the parentheses are not written, as in .Alternative notation includes \rho (\Phi) from and . Main definitions In this section, we give some definitions of the rank of a matrix. Many definitions are possible; see Alternative definitions for several of these. The column rank of is the dimension of the column space of , while the row rank of is the dimension of the row space of . A fundamental result in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Almost Certainly
In probability theory, an event (probability theory), event is said to happen almost surely (sometimes abbreviated as a.s.) if it happens with probability 1 (or Lebesgue measure 1). In other words, the set of possible exceptions may be non-empty, but it has probability 0. The concept is analogous to the concept of "almost everywhere" in measure theory. In probability experiments on a finite sample space, there is no difference between ''almost surely'' and ''surely'' (since having a probability of 1 often entails including all the sample points). However, this distinction becomes important when the sample space is an infinite set, because an infinite set can have non-empty subsets of probability 0. Some examples of the use of this concept include the strong and uniform versions of the law of large numbers, and the continuity of the paths of Brownian motion. The terms almost certainly (a.c.) and almost always (a.a.) are also used. Almost never describes the opposite of ''almost ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Coordinates
In geometry, line coordinates are used to specify the position of a line just as point coordinates (or simply coordinates) are used to specify the position of a point. Lines in the plane There are several possible ways to specify the position of a line in the plane. A simple way is by the pair where the equation of the line is ''y'' = ''mx'' + ''b''. Here ''m'' is the slope and ''b'' is the ''y''-intercept. This system specifies coordinates for all lines that are not vertical. However, it is more common and simpler algebraically to use coordinates where the equation of the line is ''lx'' + ''my'' + 1 = 0. This system specifies coordinates for all lines except those that pass through the origin. The geometrical interpretations of ''l'' and ''m'' are the negative reciprocals of the ''x'' and ''y''-intercept respectively. The exclusion of lines passing through the origin can be resolved by using a system of three coordinates to specif ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homogeneous Coordinates
In mathematics, homogeneous coordinates or projective coordinates, introduced by August Ferdinand Möbius in his 1827 work , are a system of coordinates used in projective geometry, just as Cartesian coordinates are used in Euclidean geometry. They have the advantage that the coordinates of points, including points at infinity, can be represented using finite coordinates. Formulas involving homogeneous coordinates are often simpler and more symmetric than their Cartesian counterparts. Homogeneous coordinates have a range of applications, including computer graphics and 3D computer vision, where they allow affine transformations and, in general, projective transformations to be easily represented by a matrix. If homogeneous coordinates of a point are multiplied by a non-zero scalar then the resulting coordinates represent the same point. Since homogeneous coordinates are also given to points at infinity, the number of coordinates required to allow this extension is one more than ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bézier Curve
A Bézier curve ( ) is a parametric curve used in computer graphics and related fields. A set of discrete "control points" defines a smooth, continuous curve by means of a formula. Usually the curve is intended to approximate a real-world shape that otherwise has no mathematical representation or whose representation is unknown or too complicated. The Bézier curve is named after French engineer Pierre Bézier (1910–1999), who used it in the 1960s for designing curves for the bodywork of Renault cars. Other uses include the design of computer fonts and animation. Bézier curves can be combined to form a Bézier spline, or generalized to higher dimensions to form Bézier surfaces. The Bézier triangle is a special case of the latter. In vector graphics, Bézier curves are used to model smooth curves that can be scaled indefinitely. "Paths", as they are commonly referred to in image manipulation programs, are combinations of linked Bézier curves. Paths are not bound by the limi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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