Talk:Spline (mathematics)

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[edit] Bezier curve not a spline ?

I did a complete rewrite of the article. A Bezier curve is not a spline. MathMartin 18:02, 19 Sep 2004 (UTC)

What is the difference? 20 Mar 05

To explain what a spline is I think it is best to contrast splines with polynomials. Splines are then defined as piecewiese polynomials (of course you can consider a polynomial a spline with only one piece). After this difference is clear you can discuss different forms of the polynomials used to construct the spline (like Bernstein form, Hermite form, Monomial form etc.).

So although in some sense a Bezier curve is a spline I think it is clearer not use it as a central example in the main spline article. MathMartin 16:04, 21 Mar 2005 (UTC)

I strongly agree. A Bezier curve may be a smaller portion of a spline, and one might argue it is a spline made of one curve, but it is a horrible example for the article 'Spline'. Mgsloan 23:13, 15 March 2007 (UTC)

[edit] History section

The greater part of the section History is copied verbatim from a post to the NA Digest. Can anybody confirm that we have permission to do this? I also asked the anonymous contributor for clarification at User talk:205.250.40.97. -- Jitse Niesen (talk) 16:16, 6 October 2005 (UTC)

Resolved after some e-mails. -- Jitse Niesen (talk) 12:51, 18 October 2005 (UTC)

[edit] This page needs images!

TERdON


I have an Excel spreadsheet that calculates a cubic spline, and it produces a graph showing a comparison of the cubic spline and a straight-line. I could always scan this image, but if anyone knows a better way to do something with this spreadsheet to produce a good clear image, I am glad to send you my file or to try it myself. Note: I have very little computer skills - if you want me to do something, you'll have to tell me step by step!

Sarum blue 14:58, 2 February 2006 (UTC) Sarum Blue

[edit] Definition: Closed subintervals vs. Half-open subintervals

In the Definition, some people want to use closed subintervals [ti,ti + 1] of [a,b] (option 1), while others want to use half-open subintervals [ti,ti + 1) of [a,b] (option 2a) or of [a,b) (option 2b).

The differences between these options are NOT merely cosmetic! It is important to note why (1) is so mathematically different from (2a) and (2b). Under (1), any two consecutive subintervals will share a knot, so they are not disjoint. In other words, all subintervals (together) do not constitute a partition of [a,b], but only a covering (and not a packing). All this implies that two neighboring polynomial pieces will, under (1), always match continuously over the knot in common. This will exclude step functions, for example, or any spline with discontinuities for that matter.

If we are to allow discontinuous splines, we are motivated to use half-open subintervals. Option (2b) would then be the most rigorous, while (2a) offers an attractive compromise between (1) and (2b).

[edit] Abstract: Subject Classification

The abstract used to start with "In the mathematical subfield of numerical analysis, ...", and "In the computer science subfields of computer-aided design and computer graphics, ...".

Apparently this formulation was not clear to all, since an editor [07:55, 18 October 2006 211.29.178.155] changed "subfield" to "field" since it "Doesn't make sense to talk of a *sub*field without mentioning a larger encompassing field!".

Since I contributed the original formulation, let me try to clarify what I intended by it. I thought it was clear that the "encompassing fields" were "mathematics" on the one hand and "computer science" on the other, while the SUBfields were "numerical analysis" on the one hand, and "computer-aided design" and "computer graphics" on the other. If anyone knows how to formulate this more clearly, please feel free to post your proposition.

For your information, these (sub)fields were taken from the 2000 Mathematical Subject Classification (MSC2000) of the American Mathematical Society (AMS): See http://www.ams.org/msc/

[edit] Definition: Knots are they points, values, vertices, vectors, or ... ?

Since this question was raised by an editor [09:41, 28 September 2006 SpaceDude (knots are not points, they are scalar values? would like confirmation on this... how can inequality be used on points?)], let me try to provide a short answer.

In the article's definition of splines, knots are elements of \mathbb{R}. Whether you call them "points", "vectors", "values", or ... depends on what you mean by \mathbb{R} in the first place.

If you equip the set \mathbb{R} with the usual structure of an affine space, you can call the knots "points", to emphasize the geometric view of \mathbb{R} as a manifold. You can then consider "weighted averages" of knots. Such averages are needed by the "de Boor algorithm".

If you equip the set \mathbb{R} with the usual real vector space structure, you can call the knots "vectors", to emphasize the differential-geometric view of \mathbb{R} as a tangent space/line. You can then consider "sums" and "real multiples" of knots. Very useful for uniform partitions and Fourier techniques to construct B-splines ("box splines").

If you equip the set \mathbb{R} with the usual structure of a field, you can call the knots "values", to emphasize the algebraic view of \mathbb{R} as a space of numbers. You can then consider "sums", "differences", "products" and "quotients" of knots. Very useful for the divided difference approach to B-splines.

If you equip the set \mathbb{R} with the usual ordering structure, you could call the knots "vertices" or "exposed points", to empasize the convexity of the subintervals. You can then consider "in-betweenness" of knots and whether any knot is larger than another one. Needed by polyhedron projection techniques to construct B-splines geometrically ("polyhedral splines", "box splines", "simplex splines", "cone splines").

And so on ...

Now, the set \mathbb{R} is frequently assumed to be equiped with SEVERAL such structures at the same time -- this is not contradictory as long as the structures are compatibile with each other (which they are), so they can be combined meaningfully. This is how "inequalities" can be used on "points": take \mathbb{R} with the usual affine structure and the usual ordering structure combined.

But the assumed (combined) structure(s) are not always stated explicitly though. Fortunately, it is not difficult to work backwards and recover the needed structure(s) on \mathbb{R} from the way the elements of \mathbb{R} are used.

So what are they then, these knots. Are they points? vectors? values? vertices? ... It depends on the writer.

My impression is that computer scientists prefer to maximally equip \mathbb{R} with all structures possible, so they can do what they want with its elements (without having to bother about what they mean by \mathbb{R}) and can call them how they want: "vectors", "values", "points", ... (Similarly, in "3D" they can speak of "points" and "vectors" -- although CAGD writers definitely prefer "vectors").

Mathematicians, out of economy and clarity, would equip \mathbb{R} only minimally: they wouldn't give it a particular structure unless it is really needed.

For example, in CAGD text books one speaks of "vectors" even if the origin has no special role, so (affine) "points" would do for mathematicians.

To conclude, knots in \mathbb{R} can be called "vectors", "values", "points", "vertices", ...

Nevertheless, the term "value" has the disadvantage that it is (so far) not used in multivariate spline theory (not discussed here).

[edit] examples

Hi the example is ok but need to explicitly show the smoothness vector \mathbf r. SNx 20:11, 4 December 2006 (UTC)

[edit] Spline smoothness

In the example section, it is said that second derivative should be set to 0 at end points so that the spline be a straight line outside of the interval, while not disrupting its smoothness. This forces the spline to be straight at end points, but to force smoothness, one need to get the same slope on both sides of the end points, so S'_{1}(a) \, = S'_{2}(a) where S1 and S2 stands for the two polynomials each side of a. 132.246.2.25 21:32, 23 July 2007 (UTC)Matthieu Deveau

[edit] Polynomials are in C\infty

Expressing the connectivity as a "loss of smoothness" is reasonable, since if S were a simple polynomial throughout a neighborhood of ti, it would have smoothness Cn at ti, and you would expect to lose smoothness in order to break a polynomial apart into pieces.

Every polynomial is in C^{\infty} so it would have smoothness C^{\infty} at ti.

However, it's a fact that for 2 polynomials of degree n it's enough their 0 to n derivatives are equal at point p, for these polynomials to be the same one poly. in fact (hence, to be C^{\infty} smooth at point p). Eventually, I understand the idea behind n-"number of knots" loss of smoothness, but it was a bit misleading at first. Any ideas how to improve that? SalvNaut 21:03, 27 September 2007 (UTC)