# Ways to introduce B-splines

I asked this on overflow, but it hasn't gotten many responses so I'll try here as well.

I have the option of mentoring some undergrads in a topic lying within approximation theory and I really want to do $$B$$-splines. Mostly because I have recently found applications of them in my own research and I think it's a good opportunity for me to further learn the material. (And to show them cool stuff as well of course.)

Suppose we are given a sequence of knots t $$= (t_i)_{i \in \mathbb{Z}} \subset \mathbb{R}$$. I am aware of two ways in which $$B$$-splines can be defined.

Method 1: First define the $$B$$-splines of order $$1$$ (or degree $$0$$) to be the characteristic functions $$B_{i1} = \chi_{[t_i,t_{i+1})}$$. Then we define the $$B$$-splines of higher order by the recurrence relation

$$B_{ik} = \lambda_{ik}B_{i,k-1} + (1 - \lambda_{i+1,k})B_{i+1,k-1}$$
where $$\begin{equation*} \lambda_{ik}(t) = \left\{ \begin{array}{ll} \frac{t - t_i}{t_{i+k-1} - t_i} & \quad \text{if} \ \ \ t_i \neq t_{i+k-1} \\ 0 & \quad \text{otherwise} \end{array} \right. \end{equation*}$$

I understand that this is a computationally practical way of defining $$B$$-splines and that many of the early theorems about $$B$$-splines have simple proof when given this definition. However, I am of the opinion that you wouldn't introduce $$B$$-splines this way unless you really want to bore your audience as this recurrence relation is highly unmotivated and, until you begin to actually prove theorems with it, it simply doesn't look interesting.

The next way is longer but the idea is more natural (at first at least). I don't want to make this post too long so I will skip details. I include some details for completion, but I suspect someone with an answer to this post is likely familiar with everything I mention below.

Method 2: Suppose we are investigating the problem of finding a basis for the space of piece-wise polynomials of order $$k$$ (or degree $$k-1$$) with breakpoints at $$(t_i)$$ with some specified smoothness condition at each $$t_i$$. To find this basis we first make the problem easier by finding a basis for the space of piece-wise polynomials on $$\mathbb{R}$$ with a finite set of breakpoints and some specified smoothness condition at these breakpoints. If this finite knot sequence is $$\{x_1, \dots x_n\}$$, we get led to the truncated power basis $$\{(t - x_i)_+^{j} | 1 \leq i \leq n , 0 \leq j \leq k-1\}$$.

We then find some linear combination of these truncated powers to begin constructing compactly supported piece-wise polynomials supported on closed intervals with end points belonging to our knot sequence t. (I have skipped many details here). But in order to actually define the $$B$$-splines, we need to find out the coefficients of the truncated powers that yield them. This takes us to the divided difference operators and I am not a fan of these operators either. I also find considering them to be somewhat unmotivated (albeit not as unmotivated as the first idea).

My question: Are there other ways to introduce $$B$$-splines aside from the $$2$$ methods I have given?

I suspect the solution to my dilemna is to understand these divided difference operators more in depth, but I want to know if there are other ways. I've been reading through a book on Box Splines which are defined as distributions. It seems interesting, but I've only begun and don't yet fully see how they generalize $$B$$-splines. Even if this approach would work as well, I am unsure whether it would be accessible to undergrads.

• This question may well be suited for the mathematics educators stack exchange Commented Jul 31, 2021 at 15:23
• @FShrike I think that here is a better place than on Mathematics educators stack exchange. Commented Jul 31, 2021 at 15:28
• Have you already met blossom-splines ? Commented Jul 31, 2021 at 16:11
• @FShrike I don't think it's well suited there. At some level I am asking a question about how I should teach something. However, what I'm really looking for is a motivation for splines and theorems about splines that I am not familiar with and my hope is that these theorems will guide how I introduce the material. Commented Aug 1, 2021 at 4:04

I wouldn't say that the recurrence formula is unmotivated. What's mysterious about it is the choice of the coefficients $$\lambda_{ik}$$. But, you can show that these coefficients are the only ones that will work if you want compact supports and the magical continuity properties of b-splines. For further motivation, you can look at this book draft.

I second what @Jean-Marie said: blossoming is the best way to teach people about Bezier curves and b-splines. Ramshaw's original document is overly general and hard to read, in my view, but you can find numerous simpler accounts. For example, this book by Gallier is pretty good, in my opinion.

Even of you don't get into blossoming, I'd be inclined to define b-spline curves as the things produced by the deBoor algorithm applied to a given sequence of control points.

Divided differences were the original approach to b-splines, but most people steer clear of them, these days. They're the most boring and obscure approach, in my view.

• I am very curious to see how the recurrence formula is motivated. In presentations I've seen by Carl deBoor, the recurrence relation is proven with divided differences then the recurrence relation is used to prove all the good stuff. In other treatments (also by Carl deBoor), the recurrence relation is used as the definition of B-splines, but they seem to come out of nowhere. I will skim through what you've linked, but that's the exact sort of thing I am curious about. Commented Aug 1, 2021 at 23:26
• Suppose you want to fabricate a piecewise quadratic as an affine combination of two piecewise linear "hat" functions. If you want the piecewise quadratic to be C1 continuous, there's only one affine combination that works. And it's somewhat remarkable that you can combine two functions that are not C1 and get one that is -- this is the magic of b-splines. I wrote an answer here about this, but now I can't find it. Commented Aug 2, 2021 at 1:00
• And the answer to your question "is there another way" is "yes". Blossoming. Commented Aug 2, 2021 at 1:02
• Carl deBoor is an approximation theory guy. If you want intuitive, motivated, geometric, interesting, then you have to read the CAGD crowd: see books by Farin, Gallier, Goldman, de Casteljau, etc. Commented Aug 2, 2021 at 1:06
• Found my old answer. Here: math.stackexchange.com/questions/682929/… Commented Aug 2, 2021 at 1:16