De Boor's algorithm: Difference between revisions

Revision as of 21:04, 20 September 2005

Revision as of 20:56, 3 October 2005

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We can express the curve as

:<math> \vec{s}(x) = \sum_{i=0}^{p-1} \vec{d}_i N_i^n(x) , </math>
where
<math>N_i^r(x)=\frac{x-u_i}{u_{i+n}-u_i}N_i^{r-1}(x) and- \frac{x-u_{i+n+1}}{u_{i+n+1}-u_{i+1}}N_{i+1}^{r-1}(x) ,</math>
and
<math>N_i^0(x)=\left\{\begin{matrix} 1, & \mbox{if }x \in [u_{\ell},u_{\ell+1}) \\ 0, & \mbox{otherwise } \end{matrix}\right.</math>

where ''N_iⁿ(x)'' are polynomials in x with coefficients depending on ''u₀, ..., u_p'' but not <math>\vec{d}_i</math>.

Due to the spline locality property,