Talk:B-spline

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.)

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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)

The definition of B-spline at the top, as a function into the plane, is not consistent with most of the rest of the article, where B-splines are real-valued. The definition doesn't need to be the most general, but should cover common cases at least, so whoever fixes the other points, could you have a look at this, too? Possibly a simple definition with references for variants? Reidhuntsinger (talk) 20:53, 29 April 2010 (UTC)

OK, I misunderstood. Maybe distinguish between a "B-spline curve" (the first instance of B-spline) and a B-spline? Note that the former is often referred to as a B-spline, too. Define a spline as a linear combination of B-splines, then point out that a "B-spline curve" is just a pair of splines t -> (f(t),g(t)). Reidhuntsinger (talk) 21:03, 29 April 2010 (UTC)

Blending Function

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

Changing the degree

I am not an expert on this - but the degree in the Cox-deBoor recursion looks out by 1. I will change (change back if you object) Michael Kemp. —Preceding unsigned comment added by 137.166.4.130 (talk) 00:37, 26 September 2008 (UTC)

Uniform quadratic B-spline

I think there is some information missing to make this comprehensible for people that (like me) are new to B-splines:

$b_{j,2}(t)={\begin{cases}{\frac {1}{2}}t^{2}\\-t^{2}+t+{\frac {1}{2}}\\{\frac {1}{2}}(1-t)^{2}\end{cases}}$

After quite some time of thinking I think to have found out that the parameter t is $\in [0,1]$ for all three segments, while in the form that constant and linear B-splines are given in, t does not "jump" from 1 to 0 at the segment borders. Am I right? If so, this should definitely be clarified. --Zupftom (talk) 12:36, 1 May 2009 (UTC)

P.S.: It is said that the B-spline given in the above form is uniform, but this doesn't say anything about the distance between two knots, right? —Preceding unsigned comment added by Zupftom (talk • contribs) 12:43, 1 May 2009 (UTC)

How useful would a higher-order uniform B-spline be?

D. Salomon's book "Computer graphics and geometric modeling" has some higher-order basis matrices worked out...

For example, the quartic spline:

${\cfrac {1}{4!}}{\begin{bmatrix}1&-4&6&-4&1\\-4&12&-12&4&0\\6&-6&-6&6&0\\-4&-12&12&4&0\\1&11&11&1&0\end{bmatrix}}$

Any value in putting this on the article (or at least referring to the book) ?

--Immer in Bewegung (talk) 06:06, 21 May 2009 (UTC)

IMO I wouldn't copy the matrix into the article. But a mention that quartic splines are used in computer graphics (or wording supported by that book), with that ref given, might be a good addition to the article. After all, nearly all splines used are cubic (or less).

Propose this be split into two articles

I propose that this article be split into two articles, one pertaining to the use of the term "B-spline" to refer to a curve, the other pertaining to the use of "B-spline" to refer to a mathematical function.

"Spline" already has a disambiguation page. It seems to me that "B-Spline" needs one too, for parallel reasons. Dratman (talk) 21:28, 24 January 2011 (UTC)

Has there been a discussion of a problem that this is a solution to? I don't see it. I don't quite get the distinction you're making; can you explain and point out sources for the different interpretations? Dicklyon (talk) 21:39, 24 January 2011 (UTC)

B-spline name

Isn't B in B-spline just from name Bazier. It is just like linear, quadratic, qubic splines have in each interval linear quadratic or qubic form, similary in B(azier)-splines in each subinterval we have a Bezier function (which is a function of basis polynomials aka Bernstein basis functions). So who and why, call B-splines a "basis splines" ? I do not get it, and I'm not sure if this is correct.