Splines

Synonyms

Piecewise polynomial

Related Concepts

Algebraic Curve; Parametric Curve

Definition

In mathematics, splines are piecewise continuous functions, such as polynomials, defined in successive subintervals. They are often used to represent one- or multidimensional data set (e.g., a curve or a surface) in the applications requiring interpolation, smoothing or nonrigid transformation [1]. For example, a spline curve is a piecewise collection of curve segments defined in polynomials that are connected end to end to form a single continuous curve. A curve in L-dimensional space can be simply defined by the following form:

$$S:[a,b]\rightarrow {\mathbb R}^L,$$

(1)

where function S takes variables from an interval [a,b] and maps them to an L-dimensional real number. If the interval [a,b] is divided into k ordered disjoint subintervals t _i , t_i+1 with

$$a=t_0 \leq t_1 \leq \ldots \leq t_k = b, i=0,\ldots, k-1, $$

(2)

then in each subinterval [t _i , t_i+1] there is a polynomial defined as

$$P_...