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Variation Diminishing Property Of Bézier Curves
In mathematics, the variation diminishing property of certain mathematical objects involves diminishing the number of changes in sign (positive to negative or vice versa). Variation diminishing property for Bézier curves The variation diminishing property of Bézier curves is that they are smoother than the polygon formed by their control points. If a line is drawn through the curve, the number of intersections with the curve will be less than or equal to the number of intersections with the control polygon. In other words, for a Bézier curve ''B'' defined by the control polygon P, the curve will have no more intersection with any plane as that plane has with P. This may be generalised into higher dimensions. This property was first studied by Isaac Jacob Schoenberg in his 1930 paper, . He went on to derive it by a transformation of Descartes' rule of signs. Proof The proof uses the process of repeated degree elevation of Bézier curve. The process of degree elevation f ... [...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] |
Isaac Jacob Schoenberg
Isaac Jacob Schoenberg (April 21, 1903 – February 21, 1990) was a Romanian-American mathematician, known for his invention of splines. Life and career Schoenberg was born in Galați. He studied at the University of Iași, receiving his M.A. in 1922. From 1922 to 1925 he studied at the Universities of Berlin and Göttingen, working on a topic in analytic number theory suggested by Issai Schur. He presented his thesis to the University of Iași, obtaining his Ph.D. in 1926. In Göttingen, he met Edmund Landau, who arranged a visit for Schoenberg to the Hebrew University of Jerusalem in 1928. During this visit, Schoenberg began his work on total positivity and variation-diminishing linear transformations. In 1930, he returned from Jerusalem, and married Landau's daughter Charlotte in Berlin. In 1930, he was awarded a Rockefeller Fellowship, which enabled him to go to the United States, visiting the University of Chicago, Harvard, and the Institute for Advanced St ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Descartes' Rule Of Signs
In mathematics, Descartes' rule of signs, first described by René Descartes in his work ''La Géométrie'', is a technique for getting information on the number of positive real roots of a polynomial. It asserts that the number of positive roots is at most the number of sign changes in the sequence of polynomial's coefficients (omitting the zero coefficients), and that the difference between these two numbers is always even. This implies, in particular, that if the number of sign changes is zero or one, then there are exactly zero or one positive roots, respectively. By a homographic transformation of the variable, one may use Descartes' rule of signs for getting a similar information on the number of roots in any interval. This is the basic idea of Budan's theorem and Budan–Fourier theorem. By repeating the division of an interval into two intervals, one gets eventually a list of disjoint intervals containing together all real roots of the polynomial, and containing each exactly ... [...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 (x_1,y_1), the linear interpolant is the straight line between these points. For a value in the interval (x_0, x_1), the value 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 , which is the unknown value at , gives \begin y &= y_0 + (x-x_0)\frac \\ &= \frac + \frac\\ &= \frac \\ &= \frac, \end which is the formula for linear interpolation in the interval (x_0,x_1). Outside this interval, the formula is identical to linear extrapolation. This formula can also be understood as a weighted average. The weights are inv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elsevier
Elsevier () is a Dutch academic publishing company specializing in scientific, technical, and medical content. Its products include journals such as ''The Lancet'', ''Cell'', the ScienceDirect collection of electronic journals, '' Trends'', the '' Current Opinion'' series, the online citation database Scopus, the SciVal tool for measuring research performance, the ClinicalKey search engine for clinicians, and the ClinicalPath evidence-based cancer care service. Elsevier's products and services also include digital tools for data management, instruction, research analytics and assessment. Elsevier is part of the RELX Group (known until 2015 as Reed Elsevier), a publicly traded company. According to RELX reports, in 2021 Elsevier published more than 600,000 articles annually in over 2,700 journals; as of 2018 its archives contained over 17 million documents and 40,000 e-books, with over one billion annual downloads. Researchers have criticized Elsevier for its high profit marg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totally Positive Matrix
In mathematics, a totally positive matrix is a square matrix in which all the minors are positive: that is, the determinant of every square submatrix is a positive number. A totally positive matrix has all entries positive, so it is also a positive matrix; and it has all principal minors positive (and positive eigenvalues). A symmetric totally positive matrix is therefore also positive-definite. A totally non-negative matrix is defined similarly, except that all the minors must be non-negative (positive or zero). Some authors use "totally positive" to include all totally non-negative matrices. Definition Let \mathbf = (A_)_ be an ''n'' × ''n'' matrix. Consider any p\in\ and any ''p'' × ''p'' submatrix of the form \mathbf = (A_)_ where: : 1\le i_1 < \ldots < i_p \le n,\qquad 1\le j_1 <\ldots < j_p \le n. Then A is a totally positive matrix if: [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobi Operator
A Jacobi operator, also known as Jacobi matrix, is a symmetric linear operator acting on sequences which is given by an infinite tridiagonal matrix. It is commonly used to specify systems of orthonormal polynomials over a finite, positive Borel measure. This operator is named after Carl Gustav Jacob Jacobi. The name derives from a theorem from Jacobi, dating to 1848, stating that every symmetric matrix over a principal ideal domain is congruent to a tridiagonal matrix. Self-adjoint Jacobi operators The most important case is the one of self-adjoint Jacobi operators acting on the Hilbert space of square summable sequences over the positive integers \ell^2(\mathbb). In this case it is given by :Jf_0 = a_0 f_1 + b_0 f_0, \quad Jf_n = a_n f_ + b_n f_n + a_ f_, \quad n>0, where the coefficients are assumed to satisfy :a_n >0, \quad b_n \in \mathbb. The operator will be bounded if and only if the coefficients are bounded. There are close connections with the theory of orthog ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss–Jordan Triangulation
In mathematics, Gaussian elimination, also known as row reduction, is an algorithm for solving systems of linear equations. It consists of a sequence of operations performed on the corresponding matrix of coefficients. This method can also be used to compute the rank of a matrix, the determinant of a square matrix, and the inverse of an invertible matrix. The method is named after Carl Friedrich Gauss (1777–1855) although some special cases of the method—albeit presented without proof—were known to Chinese mathematicians as early as circa 179 AD. To perform row reduction on a matrix, one uses a sequence of elementary row operations to modify the matrix until the lower left-hand corner of the matrix is filled with zeros, as much as possible. There are three types of elementary row operations: * Swapping two rows, * Multiplying a row by a nonzero number, * Adding a multiple of one row to another row. (subtraction can be achieved by multiplying one row with -1 and adding ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Planar Graph
In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points. Every graph that can be drawn on a plane can be drawn on the sphere as well, and vice versa, by means of stereographic projection. Plane graphs can be encoded by combinatorial maps or rotation systems. An equivalence class of topologically equivalent drawings on the sphere, usually with additional assumptions such as the absence of isthmuses, is called a pl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curves
A curve is a geometrical object in mathematics. Curve(s) may also refer to: Arts, entertainment, and media Music * Curve (band), an English alternative rock music group * ''Curve'' (album), a 2012 album by Our Lady Peace * "Curve" (song), a 2017 song by Gucci Mane featuring The Weeknd * ''Curve'', a 2001 album by Doc Walker * "Curve", a song by John Petrucci from ''Suspended Animation'', 2005 * "Curve", a song by Cam'ron from the album '' Crime Pays'', 2009 Periodicals * ''Curve'' (design magazine), an industrial design magazine * ''Curve'' (magazine), a U.S. lesbian magazine Other uses in arts, entertainment, and media * ''Curve'' (film), a 2015 film * BBC Two 'Curve' idents, various animations based around a curve motif Brands and enterprises *Curve (payment card), a payment card that aggregates multiple payment cards * Curve (theatre), a theatre in Leicester, United Kingdom * Curve, fragrance by Liz Claiborne * BlackBerry Curve, a series of phones from Research in Motio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
