De Boor's algorithm: Difference between revisions

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== Local support ==

B-splines have local support, meaning that the polynomials are positive only in a finitecompact ___domain and zero elsewhere. The Cox-de Boor recursion formula<ref>C. de Boor, p. 90</ref> shows this:

<math display="block">B_{i,0}(x) :=

\begin{cases}

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<math display="block"> \mathbf{S}(x) = \sum_{i=k-p}^{k} \mathbf{c}_i B_{i,p}(x). </math>

It follows from <math> i \geq 0 </math> that <math> k \geq p </math>. Similarly, we see in the recursion that the highest queried knot ___location is at index <math> k + 1 + p </math>. This means that any knot interval <math> [t_k,t_{k+1}) </math> which is actually used must have at least <math> p </math> additional knots before and after. In a [[computer program]], this is typically achieved by repeating the first and last used knot ___location <math> p </math> times. For example, for <math> p = 3 </math> and real knot locations <math> (0, 1, 2) </math>, one would pad the knot vector to <math> (0,0,0,0,1,2,2,2,2) </math>.

== The algorithm ==

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== See also ==

* [[De Casteljau's algorithm]]

* [[Bézier curve]]

* [[Non-uniform rational B-spline]]

== External links ==

* [http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/spline/de-Boor.html De Boor's Algorithm]

* [http://www.idav.ucdavis.edu/education/CAGDNotes/Deboor-Cox-Calculation/Deboor-Cox-Calculation.html The DeBoor-Cox Calculation]

== Computer code ==

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* [https://www.gnu.org/software/gsl/ GNU Scientific Library]: C-library, contains a sub-library for splines ported from [[Netlib|PPPACK]]

* [https://www.scipy.org/ SciPy]: Python-library, contains a sub-library ''scipy.interpolate'' with spline functions based on [[Netlib|FITPACK]]

* [https://github.com/msteinbeck/tinyspline TinySpline]: C-library for splines with a C++ wrapper and bindings for C#, Java, Lua, [[PHP]], Python, and Ruby

* [httphttps://einspline.sourceforge.net/ Einspline]: C-library for splines in 1, 2, and 3 dimensions with Fortran wrappers

== References ==

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'''Works cited'''

* {{cite book | author = Carl de Boor | title = A Practical Guide to Splines, Revised Edition | publisher = Springer-Verlag | year = 2003|isbn=0-387-95366-3}}

[[Category:Articles with example Python (programming language) code]]

[[Category:Numerical analysis]]

[[Category:Splines (mathematics)]]