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De Boor's algorithm
In the mathematical subfield of numerical analysis the de Boor's algorithm is a fast and numerically stable algorithm for evaluating spline curves in B-spline form. It is a generalization of the de Casteljau's algorithm for Bézier curves. The algorithm was devised by Carl R. de Boor. Simplified, potentially faster variants of the de Boor algorithm have been created but they suffer from comparatively lower stability.
== Introduction ==
The general setting is as follows. We would like to construct a curve whose shape is described by a sequence of ''p'' points , which plays the role of a ''control polygon''. The curve can be described as a function of one parameter ''x''. To pass through the sequence of points, the curve must satisfy . But this is not quite the case: in general we are satisfied that the curve "approximates" the control polygon. We assume that ''u0, ..., up-1'' are given to us along with .
One approach to solve this problem is by splines. A spline is a curve that is a piecewise ''nth'' degree polynomial. This means that, on any interval ''[ui, ui+1)'', the curve must be equal to a polynomial of degree at most ''n''. It may be equal to different polynomials on different intervals. The polynomials must be ''synchronized'': when the polynomials from intervals ''[ui-1, ui)'' and ''[ui, ui+1)'' meet at the point ''ui'', they must have the same value at this point and their derivatives must be equal (to ensure that the curve is smooth).
De Boor's algorithm is an algorithm which, given ''u0, ..., up-1'' and , finds the value of spline curve at a point ''x''. It uses O(n2) + O(n + p) operations where ''n'' is the degree and ''p'' the number of control points of ''s''.
抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』
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