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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 ...
The mathematical basis for Bézier curves—the Bernstein polynomials—was established in 1912, but the polynomials were not applied to graphics until some 50 years later when mathematician Paul de Casteljau in 1959 developed de Casteljau's algorithm, a numerically stable method for evaluating the curves, and became the first to apply them to computer-aided design at French automaker Citroën ...
The simplest spline has degree 0. It is also called a step function. The next most simple spline has degree 1. It is also called a linear spline. A closed linear spline (i.e, the first knot and the last are the same) in the plane is just a polygon. A common spline is the natural cubic spline.
Polyharmonic spline (the thin-plate-spline is a special case of a polyharmonic spline) Radial basis function (Polyharmonic splines are a special case of radial basis functions with low degree polynomial terms) Least-squares spline; Natural neighbour interpolation; Gridding is the process of converting irregularly spaced data to a regular grid ...
Lagrange and other interpolation at equally spaced points, as in the example above, yield a polynomial oscillating above and below the true function. This behaviour tends to grow with the number of points, leading to a divergence known as Runge's phenomenon ; the problem may be eliminated by choosing interpolation points at Chebyshev nodes .
A Lozenge diagram is a diagram that is used to describe different interpolation formulas that can be constructed for a given data set. A line starting on the left edge and tracing across the diagram to the right can be used to represent an interpolation formula if the following rules are followed: [5]
The simplest interpolation method is to locate the nearest data value, and assign the same value. In simple problems, this method is unlikely to be used, as linear interpolation (see below) is almost as easy, but in higher-dimensional multivariate interpolation, this could be a favourable choice for its speed and simplicity.
Polynomial curves fitting points generated with a sine function. The black dotted line is the "true" data, the red line is a first degree polynomial, the green line is second degree, the orange line is third degree and the blue line is fourth degree. The first degree polynomial equation = + is a line with slope a. A line will connect any two ...