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Please discuss this issue on the article’s talk page. In computer-aided design and computer graphics, spline functions are constructed as linear combinations of B-splines with a set of control points. The term «B-spline» was coined by Isaac Jacob Schoenberg and is short for basis spline. The places where the pieces meet are known as knots. The key property of spline functions is that they and their derivatives may be continuous, depending on the multiplicities of the knots. B-splines, and there is only one unique combination for each spline function. Alternatively, B-splines can be defined by construction by means of the Cox-de Boor recursion formula.

B-spline function is a combination of flexible bands that passes through the number of points that are called control points and creates smooth curves. These functions enable the creation and management of complex shapes and surfaces using a number of points. B-spline function and B├ęzier functions are applied extensively in shape optimization methods. The B-spline contributes only in the range between the first and last of these knots and is zero elsewhere.

A B-spline is a continuous function at the knots. 1 for each additional coincident knot. B-splines may share a subset of their knots, but two B-new boor defined over exactly the same knots are identical. In other words, a B-spline is uniquely defined by its knots. One distinguishes internal knots and end points.

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B-spline which affect the internal knot intervals. The values of the endpoints do not matter, usually the first or last internal knot is just repeated. B-splines play the role of basis functions for the spline function space, hence the name. This property follows from the fact that all pieces have the same continuity properties, within their individual range of support, at the knots. The corresponding Bs are zero outside those respective ranges.

However, because B-spline basis functions have local support, B-splines are typically computed by algorithms that do not need to evaluate basis functions where they are zero, such as de Boor’s algorithm. This relation leads directly to the FORTRAN-coded algorithm BSPLV which generates values of the B-splines of order n at x. The following scheme illustrates how each piece of order n is a linear combination of the pieces of B-splines of order n-1 to its left. These pieces are shown in the diagram.

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