Talk:B-spline

message: The definiton of B-spline is not correct:
# Knots may repeat, i.e. t_i <= t_{i+1}, not t_i < t_{i+1} as article states.
# 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.
# 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.
# 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:
# The full name of Cox-de Boor formula is "Mansfield-de Boor-Cox".
# 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)
replies: text: 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)
thread_title: Old stuff

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

message: 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)
thread_title:  Changing the degree 

message: 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)
thread_title:  [[B-Spline#Uniform quadratic B-spline|Uniform quadratic B-spline]] 

message: 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)
replies: text: 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)
thread_title:  Propose this be split into two articles 

message: 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.  —Preceding unsigned comment added by 149.156.82.207 (talk) 19:13, 7 April 2011 (UTC)
replies: text: The relation of "basis spline" to Bezier curves is covered in many books. Dicklyon (talk) 20:30, 14 August 2011 (UTC)
thread_title:  B-spline name 

message: I added the external link to my free Web e-book.  It is a more complete treatment of piecewise interpolation for those new to the subject and aimed at graphics.  I'll add references to other related Wiki pages in the future. 
My e-book is both a study of the fundamentals described in simple terms as well as a reference showing many types of Polynomial Interpolation -both common types and some developed by the author.  It also shows some techniques not seen elsewhere.   Linear interpolation is looked at carefully and shown is as the basis for all more advanced types using only Algebra.  Only after understanding how adding squared and cubed terms cause smooth curves, are the more advanced curves examined such as Bezier, Catmul-Rom, b-spline, and Hermite.  More advanced mathematical concepts and notations are kept to a minimum.  Some additional techniques that are suggested by the mathematics and that the author has not seen elsewhere are examined.  A reference is also included with all the most common curve drawing methods and includes drawings to allow comparison with other types.
WHY: I had hoped, but failed to find a book explaining polynomial interpolation basics and thought that one must certainly exist with a collection of interpolation types. I found either purely mathematical 
tests or advanced graphics texts.  Several years later, I started reading the original Internet Usenet "groups", comp.graphics.algorithms, and did lots of searching on the net.  I saved whatever I found related to splines and curves, but did not look at it or try to understand any of it until early in 1996, I decided to look at what I had collected, and started to figure things out.     I begin recording my thoughts for future reference and this is the result. The very book I wanted originally is now freely available for others.    
I do not believe this has any issues with the Wiki self cite guidelines.   -- Steve --  (talk) 22:48, 28 August 2011 (UTC)
thread_title:  External links 

message: Is it true that a cardinal spline is a b spline? Just that we assume that consecutive t_i's have distance 1? I think a remark along those lines in the text would really help understand it.
12.25.180.20 (talk) 22:57, 13 June 2013 (UTC)
replies: text: This article covers the "cardinal b-spline" (which has uniform spacing like you write), but there are other splines which include the "cardinal" name. Unfortunately this is kinda confusing and I personally wouldn't mind clarifying it in the text, but many spline articles should include such clarifications the as well. 2A02:908:1866:5B00:6583:DFF9:E55:9DDE (talk) 15:29, 22 May 2018 (UTC)
thread_title:  Cardinal splines confusing 

message: I have done a major revision of this article which was rather confused. In particular, the bits pertaining to computer graphics are now in a separate section. I have had to delete statements such as
"Put in matrix-form, it is:Splitting a uniform B-spline curve
: \mathbf{S}_i(t) = \begin{bmatrix} t^2 & t & 1 \end{bmatrix} \frac{1}{2} \begin{bmatrix}
1 & -2 & 1 \\
-2 &  2 & 0 \\
1 &  1 & 0 \end{bmatrix}
\begin{bmatrix} \mathbf{p}_{i-1} \\ \mathbf{p}_{i} \\ \mathbf{p}_{i+1} \end{bmatrix}
 for t \in [0,1], i = 1,2 \ldots m-2".
because
#citing a web-site is not sufficient for verification
#there is no reference to any such formula in Non-uniform rational B-spline, though it is clearly related
#I don't know of a book which could be used to verify the formula (I'm not sufficiently knowledgeable about computer graphics).
If someone can provide an acceptable verification source, it could be re-instated into the section on computer graphics,  but it will need further explanation (e.g. uniform knots) and perhaps a revision of the terminology. Petergans (talk) 17:25, 13 August 2013 (UTC)
thread_title:  Major revision 

message: It is not clear what the domains and ranges of any functions defined in this article are. The standard notation is to denote a function f with domain A and range B by  f : A \rightarrow B . Please adhere to this.  — Preceding unsigned comment added by 128.32.92.238 (talk) 20:28, 23 September 2013 (UTC)
replies: text: The notation follows that in de Boor's book. The domain of a B-spline is clearly stated in the first sentence of Definition. BTW, non-mathematicians will not understand the notation  f : A \rightarrow B . Petergans (talk) 08:57, 24 September 2013 (UTC)
thread_title:  Definitions of functions 

message: I find the definition section highly confusing and little illuminating! The problem is that it is completely unclear why the basis sometimes has 1, sometimes 2 and sometimes 3 subscript indices. It would be great if someone could make that a bit more rigorous. (22. Jan 2014)  — Preceding unsigned comment added by 149.217.40.222 (talk) 13:54, 22 January 2014 (UTC)
thread_title:  Very confusing Definition section 

message: The article says "Whereas Bézier curves evolve into only one parametric direction, usually called s or u, NURBS surfaces evolve into two parametric directions, called s and t or u and v." This sounds like it's making the distinction between curves and surfaces. But this isn't fundamental to NURBS versus B-splines. Am I misunderstanding? You can certainly take the tensor product of B-splines or Bézier curves to get surfaces. —Ben FrantzDale (talk) 17:28, 7 May 2014 (UTC)
replies: text: I think you are right. Ossi (talk) 00:50, 10 May 2014 (UTC)
thread_title:  Dubious: number of parametric directions 

message: The WP:LEAD of this article is basically incomprehensible unless you read the "introduction" first, even though it's basically the same material and the definitions aren't that complicated. I suggest replacing most of the current lead with the material from the section called "introduction" and abolishing the latter. JMP EAX (talk) 11:58, 15 August 2014 (UTC)
thread_title:  Lead and "introduction" 

message: The much more confused article Bézier spline duplicates some of this, although this (B-spline) article isn't exactly peachy either for it fails to explain how B-splines relate to piecewise/composite Bézier curves. JMP EAX (talk) 15:06, 15 August 2014 (UTC)
replies: text: Agreed. There needs to be a combined effort on Bézier spline, Composite Bézier curve, Bézier surface, B-spline, and NURBS. —Ben FrantzDale (talk) 14:39, 16 September 2014 (UTC)
thread_title:  Duplication & confusion elsewhere, lack of info here 

message: Substituted at 08:45, 29 April 2016 (UTC)
thread_title: Assessment comment

message: The Cardinal Spline section uses the function
B_{i,n,t}(x) = \frac{x-t_i}{h} n[0,\dots,n](. - t_i)^{n-1}_+
but that subscript plus symbol is certainly not conventional notation, and needs explaining, or better: this function needs rewriting to a more conventional notation so that the majority of readers can understand it without needing to leave wikipedia to find out what the actual meaning  of the notation is. Pomax (talk) 17:41, 23 July 2016 (UTC)
thread_title:  Lack of notation explanation in Cardinal Spline section 

message: Currently, the Definition section's first paragraph gives a definition of a B-spline.
The second paragraph goes on to claim that For any given set of knots, the B-spline is unique, hence the name, B being short for Basis.
However, a (non-zero) multiple of a B-spline is again a B-spline for the same knots with the current definition, so uniqueness does not hold.
-- 66.253.183.240 (talk) 17:54, 9 October 2016 (UTC)
thread_title:  Uniqueness of B-splines does not follow from definition 

message: Hello fellow Wikipedians,
I have just modified  on B-spline. Please take a moment to review my edit. If you have any questions, or need the bot to ignore the links, or the page altogether, please visit this simple FaQ for additional information. I made the following changes:
*Added archive https://web.archive.org/web/20070531044251/http://math.fullerton.edu:80/mathews/n2003/B-SplinesMod.html to http://math.fullerton.edu/mathews/n2003/B-SplinesMod.html
When you have finished reviewing my changes, please set the checked parameter below to true or failed to let others know (documentation at ).
Cheers.—InternetArchiveBot (Report bug) 05:06, 23 October 2016 (UTC)
thread_title:  External links modified 

message: I don't mean to be obnoxious here. Every write-up I've ever seen on splines immediately launches into mathematical details. It would be really helpful if this write-up started by painting a 60,000 foot view of splines to orient the reader. What are the basic ideas here? What are we trying to do and why? What's the "Spines for Dummies" story. Five years ago, I struggled through the early parts Dierckx's book. I have since forgotten. Am I now willing to spend many, many hours reading through minutiae ??? Most people need splines as part of a larger task and do not have the time and energy to read all of the gory details.  — Preceding unsigned comment added by 2601:648:8400:554:BEE9:2FFF:FE99:5FBC (talk) 15:57, 7 December 2022 (UTC)
thread_title:  Introduction 

message: It would be great to see the control points on these graphs, like on the first graph. 2A02:8071:B783:4980:B8F4:7DD9:7183:8BF0 (talk) 18:45, 5 June 2023 (UTC)
thread_title:  Cardinal spline graphics