Spline (mathematics)

In mathematics, a spline is a numeric function that is piecewise-defined by polynomial functions, and which possesses a high degree of smoothness at the places where the polynomial pieces connect (which are known as knots).

In interpolation problems, spline interpolation is often preferred to polynomial interpolation because it yields similar results to interpolating with higher degree polynomials while avoiding instability due to Runge's phenomenon. In computer graphics, parametric curves whose coordinates are given by splines are popular because of the simplicity of their construction, their ease and accuracy of evaluation, and their capacity to approximate complex shapes through curve fitting and interactive curve design.

The most commonly used splines are cubic splines, i.e., of order 3—in particular, cubic B-spline, which is equivalent to C2 continuous composite Bézier curves. They are common, in particular, in spline interpolation simulating the function of flat splines.

The term spline is adopted from the name of a flexible strip of metal commonly used by drafters to assist in drawing curved lines.

A simple example of a quadratic spline (a spline of degree 2) is

for which ${\displaystyle S'(0)=2}$.

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