De Boor's algorithm: Difference between revisions

Revision as of 22:08, 4 June 2009

Revision as of 08:15, 1 July 2009

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\mathbf{s}(u_{p-1})=\mathbf{d}_{p-1}</math>. But this is not quite the case: in general we are satisfied that the curve "approximates" the control polygon. We assume that ''u₀, ..., u_p-1'' are given to us along with <math>\mathbf{d}_0, \mathbf{d}_1, \dots, \mathbf{d}_{p-1}</math>.

One approach to solve this problem is by [[spline (mathematics)|spline]]s. A spline is a curve that is a piecewise ''n^th'' degree polynomial. This means that, on any interval ''<nowiki>[</nowiki>u_i, u_i+1)'', the curve must be equal to a polynomial of degree at most ''n''. It may be equal to a different polynomials on different intervals. The polynomials must be ''synchronized'': when the polynomials from intervals ''<nowiki>[</nowiki>u_i-1, u_i)'' and ''<nowiki>[</nowiki>u_i, u_i+1)'' meet at the point ''u_i'', 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, ..., u_p-1'' and <math>\mathbf{d}_0, \mathbf{d}_1, \dots, \mathbf{d}_{p-1}</math>, finds the value of spline curve <math>\mathbf{s}(x)</math> at a point ''x''. It uses [[Big O notation|O]](n²) operations. Notice that the running time of the algorithm depends only on degree ''n'' and not on the number of points ''p''.