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In the mathematical field of numerical analysis, interpolation is a type of estimation, a method of constructing (finding) new data points based on the range of a discrete set of known data points.
In the mathematical subfield of numerical analysis, a B-spline or basis spline is a spline function that has minimal support with respect to a given degree, smoothness, and domain partition. Any spline function of given degree can be expressed as a linear combination of B-splines of that degree. Cardinal B-splines have knots that are equidistant from each other. B-splines can be used for curve-fitting and numerical differentiation of experimental data.
In mathematics, the Hermite polynomials are a classical orthogonal polynomial sequence.
In mathematics, a spline is a function defined piecewise by polynomials. In interpolating problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results, even when using low degree polynomials, while avoiding Runge's phenomenon for higher degrees.
In the mathematical field of numerical analysis, spline interpolation is a form of interpolation where the interpolant is a special type of piecewise polynomial called a spline. That is, instead of fitting a single, high-degree polynomial to all of the values at once, spline interpolation fits low-degree polynomials to small subsets of the values, for example, fitting nine cubic polynomials between each of the pairs of ten points, instead of fitting a single degree-nine polynomial to all of them. Spline interpolation is often preferred over polynomial interpolation because the interpolation error can be made small even when using low-degree polynomials for the spline. Spline interpolation also avoids the problem of Runge's phenomenon, in which oscillation can occur between points when interpolating using high-degree polynomials.
In mathematics, a 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.
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, that is, by its values and first derivatives at the end points of the corresponding domain interval.
In numerical analysis, Hermite interpolation, named after Charles Hermite, is a method of polynomial interpolation, which generalizes Lagrange interpolation. Lagrange interpolation allows computing a polynomial of degree less than n that takes the same value at n given points as a given function. Instead, Hermite interpolation computes a polynomial of degree less than n such that the polynomial and its first few derivatives have the same values at m given points as the given function and its first few derivatives at those points. The number of pieces of information, function values and derivative values, must add up to .
In the mathematical field of numerical analysis, monotone cubic interpolation is a variant of cubic interpolation that preserves monotonicity of the data set being interpolated.
In numerical analysis, multivariate interpolation is interpolation on functions of more than one variable ; when the variates are spatial coordinates, it is also known as spatial interpolation.
Ribbon diagrams, also known as Richardson diagrams, are 3D schematic representations of protein structure and are one of the most common methods of protein depiction used today. The ribbon depicts the general course and organisation of the protein backbone in 3D and serves as a visual framework for hanging details of the entire atomic structure, such as the balls for the oxygen atoms attached to myoglobin's active site in the adjacent figure. Ribbon diagrams are generated by interpolating a smooth curve through the polypeptide backbone. α-helices are shown as coiled ribbons or thick tubes, β-sheets as arrows, and non-repetitive coils or loops as lines or thin tubes. The direction of the polypeptide chain is shown locally by the arrows, and may be indicated overall by a colour ramp along the length of the ribbon.
In mathematics, Hermite numbers are values of Hermite polynomials at zero argument. Typically they are defined for physicists' Hermite polynomials.
Charles Hermite FRS FRSE MIAS was a French mathematician who did research concerning number theory, quadratic forms, invariant theory, orthogonal polynomials, elliptic functions, and algebra.
Charles Hermite (1822–1901) was a French mathematician.
In mathematics, the Askey scheme is a way of organizing orthogonal polynomials of hypergeometric or basic hypergeometric type into a hierarchy. For the classical orthogonal polynomials discussed in Andrews & Askey (1985), the Askey scheme was first drawn by Labelle (1985) and by Askey and Wilson, and has since been extended by Koekoek & Swarttouw (1998) and Koekoek, Lesky & Swarttouw (2010) to cover basic orthogonal polynomials.
Gerlind Plonka-Hoch is a German applied mathematician specializing in signal processing and image processing, and known for her work on refinable functions and curvelets. She is a professor at the University of Göttingen, in the Institute for Numerical and Applied Mathematics.
References
- ↑ Parekh, Ranjan (2019-11-26). Fundamentals of Graphics Using MATLAB. CRC Press. p. 53. ISBN 978-0-429-59173-0.
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