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Cubic Spline
In numerical analysis, a cubic Hermite spline or cubic Hermite interpolator is a spline where each piece is a third-degree polynomial specified in Hermite form:[1] that is, by its values and first derivatives at the end points of the corresponding domain interval. Cubic Hermite splines are typically used for interpolation of numeric data specified at given argument values x 1 , x 2 , … , x n displaystyle x_ 1 ,x_ 2 ,ldots ,x_ n , to obtain a smooth continuous function. The data should consist of the desired function value and derivative at each x k displaystyle x_ k
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Hermite Polynomial
In mathematics, the Hermite polynomials
Hermite polynomials
are a classical orthogonal polynomial sequence. The polynomials arise in:probability, such as the Edgeworth series; in combinatorics, as an example of an Appell sequence, obeying the umbral calculus; in numerical analysis as Gaussian quadrature; in physics, where they give rise to the eigenstates of the quantum harmonic oscillator; in systems theory in connection with nonlinear operations on Gaussian noise. in random matrix theory in Wigner–Dyson ensembles.
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Floor Function
In mathematics and computer science, the floor function is the function that takes as input a real number x displaystyle x and gives as output the greatest integer less than or equal to x displaystyle x , denoted floor ( x ) = ⌊ x ⌋ displaystyle text floor (x)=lfloor xrfloor
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De Casteljau Algorithm
In the mathematical field of numerical analysis, De Casteljau's algorithm is a recursive method to evaluate polynomials in Bernstein form or Bézier curves, named after its inventor Paul de Casteljau
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Centripetal Catmull–Rom Spline
In computer graphics, centripetal Catmull–Rom spline is a variant form of Catmull-Rom spline
Catmull-Rom spline
[1] formulated by Edwin Catmull
Edwin Catmull
and Raphael Rom according to the work of Barry and Goldm
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Edwin Catmull
Edwin "Ed" Earl Catmull (born March 31, 1945) is an American computer scientist and current president of Pixar
Pixar
and Walt Disney
Disney
Animation Studios.[3][4][5] As a computer scientist, Catmul
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Raphael Rom
Raphael "Raphi" Rom is an Israeli computer scientist working at Technion – Israel
Israel
Institute of Technology.[1] Rom earned his Ph.D. in 1975 from the University of Utah, under the supervision of Thomas Stockham.[2] He is known for his contribution to the development of the Catmull–Rom spline,[3] and for his research on computer networks. Selected publications[edit]Catmull, E.; Rom, R. (1974), "A class of local interpolating splines", in Barnhill, R. E.; Riesenfeld, R. F., Computer Aided Geometric Design, New York: Academic Press, pp. 317–326 . Rom, R.; Sidi, M. (1990), Multiple Access Protocols: Performance and Analysis, Springer .[4][5] Orda, A.; Rom, R. (1990), "Shortest-path and minimum-delay algorithms in networks with time-dependent edge-length", Journal of the ACM, 37 (3): 607–625, doi:10.1145/79147.214078, MR 1072271 . Orda, A..; Rom, R.; Shimkin, N
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Key Frame
A keyframe in animation and filmmaking is a drawing that defines the starting and ending points of any smooth transition. The drawings are called "frames" because their position in time is measured in frames on a strip of film. A sequence of keyframes defines which movement the viewer will see, whereas the position of the keyframes on the film, video, or animation defines the timing of the movement. Because only two or three keyframes over the span of a second do not create the illusion of movement, the remaining frames are filled with inbetweens.Contents1 Use of keyframes as a means to change parameters 2 Video editing 3 Video compression 4 See also 5 ReferencesUse of keyframes as a means to change parameters[edit] In software packages that support animation, especially 3D graphics, there are many parameters that can be changed for any one object. One example of such an object is a light (In 3D graphics, lights function similarly to real-world lights
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Kochanek–Bartels Spline
In mathematics, a Kochanek–Bartels spline
Kochanek–Bartels spline
or Kochanek–Bartels curve is a cubic Hermite spline with tension, bias, and continuity parameters defined to change the behavior of the tangents. Given n + 1 knots,p0, ..., pn,to be interpolated with n cubic Hermite curve segments, for each curve we have a starting point pi and an ending point pi+1 with starting tangent di and ending tangent di+1 defined by d i = ( 1 &#
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Monotone Cubic Interpolation
In the mathematical subfield of numerical analysis, monotone cubic interpolation is a variant of cubic interpolation that preserves monotonicity of the data set being interpolated. Monotonicity is preserved by linear interpolation but not guaranteed by cubic interpolation.Contents1 Monotone cubic Hermite interpolation1.1 Interpolant selection 1.2 Cubic interpolation2 Example implementation 3 References 4 External linksMonotone cubic Hermite interpolation[edit]Example showing non-monotone cubic interpolation (in red) and monotone cubic interpolation (in blue) of a monotone data set.Monotone interpolation can be accomplished using cubic Hermite spline with the tangents m i displaystyle m_ i modified to ensure the monotonicity of the resulting Hermite spline. An algorithm is also available for monotone quintic Hermite interpolation. Interpolant selection[edit] There are several ways of selecting interpolating tangents for each data point
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Monotonic Function
In mathematics, a monotonic function[1] [2](or monotone function[3]) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory.Contents1 Monotonicity in calculus and analysis1.1 Monotonic transformation 1.2 Some basic applications and results2 Monotonicity in topology 3 Monotonicity in functional analysis 4 Monotonicity in order theory 5 Monotonicity in the context of search algorithms 6 Boolean functions 7 See also 8 Notes 9 Bibliography 10 External linksMonotonicity in calculus and analysis[edit] In calculus, a function f displaystyle f defined on a subset of the real numbers with real values is called monotonic if and only if it is either entirely non-increasing, or entirely non-decreasing.[2] That is, as per Fig
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Matrix Transpose
In linear algebra, the transpose of a matrix is an operator which flips a matrix over its diagonal, that is it switches the row and column indices of the matrix by producing another matrix denoted as AT (also written A′, Atr, tA or At)
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Numerical Analysis
Numerical analysis
Numerical analysis
is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). One of the earliest mathematical writings is a Babylonian tablet from the Yale Babylonian Collection
Yale Babylonian Collection
(YBC 7289), which gives a sexagesimal numerical approximation of the square root of 2, the length of the diagonal in a unit square. Being able to compute the sides of a triangle (and hence, being able to compute square roots) is extremely important, for instance, in astronomy, carpentry and construction.[2] Numerical analysis
Numerical analysis
continues this long tradition of practical mathematical calculations
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Tricubic Interpolation
In the mathematical subfield numerical analysis, tricubic interpolation is a method for obtaining values at arbitrary points in 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 ) = ∑ i = 0 3 ∑ j = 0 3 ∑ k = 0 3 a i j k x i y j z k . displaystyle f(x,y,z)=sum _ i=0 ^ 3 sum _ j=0 ^ 3 sum _ k=0 ^ 3 a_ ijk x^ i y^ j z^ k
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Hermite Interpolation
In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of interpolating data points as a polynomial function. The generated Hermite interpolating polynomial is closely related to the Newton polynomial, in that both are derived from the calculation of divided differences. However, the Hermite interpolating polynomial may also be computed without using divided differences, see Chinese remainder theorem § Hermite interpolation. Unlike Newton interpolation, Hermite interpolation matches an unknown function both in observed value, and the observed value of its first m derivatives
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Multivariate Interpolation
In numerical analysis, multivariate interpolation or spatial interpolation is interpolation on functions of more than one variable. The function to be interpolated is known at given points ( x i , y i , z i , … ) displaystyle (x_ i ,y_ i ,z_ i ,dots ) and the interpolation problem consist of yielding values at arbitrary points ( x , y , z , … ) displaystyle (x,y,z,dots ) . Multivariate interpolation
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