Talk:B-spline

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The definiton of B-spline is not correct:

  1. Knots may repeat, i.e. t_i <= t_{i+1}, not t_i < t_{i+1} as article states.
  2. S(t) domain is [0,1], but B-spline is defined in range [t_n, t_{m-n-1}] and thus additions are done across indexes i=0..m-n-1, not i=0..m.
  3. Since knots may repeat, denominators appear in de Boor-Cox formula, i.e. t_{i+n}-t_i and t_{i+n+1} - t_{i+1} may eqaul zero. In such cases certain part of sum just disappear.
  4. Count of control points is limited: n = m-k-1, where k is a degree of curves, and n is a max index of P.

And two notes:

  1. The full name of Cox-de Boor formula is "Mansfield-de Boor-Cox".
  2. B-spline is a shotening from "basis spline function", thus "basis B-spline" means "basis basis spline function" and it is quite redundant.

I am not native speaker, so I haven't touched the article. Someone, please consider my remarks.

The matrix form of the uniform B-splines may not be correct. If the basis formula is derived from the Bernstein polynomial, then the correct cubic matrix is [-1 3 -3 1; 3 -6 3 0; -3 3 0 0; 1 0 0 0].

An excellent reference for B-splines & Bezier curves is G. Farin's Curves and Surfaces for Computer Aided Geometric Design Academic Press, 4th ed. 1994 (I don't know the correct way to add a book reference.)

The sum should be over all points, not just 0 to "m-n-1". The reason is trivial: If the points are not included in the sum, they have no effect on the curve at any value of t. Fyo 20:20, 3 July 2007 (UTC)
Unless anyone objects, I will change the nomenclature used in this article to reflect the majority of bspline texts: n is the number of control points -1, k is the degree and m the number of knots (the article currently uses n as the degree). While p is also commonly used to denote degree, my experience is that students tend to confuse it with the capital P used for the control points. Since "k" is also used quite often in the literature, this would be my preference. Fyo 20:20, 3 July 2007 (UTC)

[edit] Blending Function

What is the exact definition of a blending function?--SiriusB 08:18, 5 September 2005 (UTC)