A couple of questions on the NURBS basis functions

Question

I read a little about NURBS curves (specifically from http://www.cs.mtu.edu/~shene/COURSES/cs3621/NOTES/), and I have a couple of questions about the motivation behind the choices made in designing the basis functions.

My current understanding is that, given input knot vector, weights, control points (say we are 1-dimensional as the generalization to higher dimensional curves or surfaces doesn't appear to be a difficulty) and degree, we have basis functions $B_{1},\ldots,B_{n}$ such that each basis function looks like a bump function and the basis functions are a partition of unity. Furthermore, if we assume that knot vector doesn't have multiplicity, then each basis function is in class $C^{d}$, where $d$ is the degree.

My questions are below:

1) The notes say that the support of $B_{i}$ is going to be $d$ intervals in the knot vector. So the support of $B_{i}$ is going to be increasing in the degree $d$. In particular, this seems to me like the curve produced is going to be less local in its dependence on the control points.

By this, I mean that if I move a control point $x_{i}$, then the curve near $x_{i-\lfloor d/2\rfloor},\ldots,x_{i+\lfloor d/2\rfloor}$ are all going to move (whatever the precisely statement is, the interval of control points grows linearly in $d$). Now, this seems like a bad thing, as I would assume we want to have the curve closely approximate the control points and making the curve depend on an interval of control points instead of the closest control point makes this harder.

However, it doesn't seem like a difficult task to make the support of $B_{i}$ smaller, say only two intervals in the knot vector, while still maintaining that it is in class $C^{d}$. I'm sure somebody has thought about this, but I can't find this easily. Why don't people do this?

2) There are examples of bump functions in basic analysis that are $C^{\infty}$. These won't be rational functions like the NURBS basis functions, but why don't we use $C^{\infty}$ bump functions as basis functions? It seems to me that when we increase the degree of the NURBS basis functions, we approximate the control points less closely so we have more regularity (differentiability). But if we use $C^{\infty}$ bump functions, we get a smooth curve.

Thanks for your time.

Answer 1

You are right -- increasing degree makes the support of the basis functions wider, so editing becomes more global, less local. This isn't necessarily a bad thing. In fact, people who design very high quality curves typically use Bezier curves (curves that have only one b-spline segment; order = number of control points). With these curves, moving a control point will move every point of the curve (except maybe the end-points). If the system lets you make local edits, then it's very easy to introduce "bumps", i.e. localized regions where the curvature is very different from what it is elsewhere. For high quality curve design, this is highly undesirable, obviously.

Why don't people use (non-polynomial) $C^{\infty}$ basis functions? Well, the basis functions used in the Bezier curves I mentioned above are $C^{\infty}$, because they are polynomials (Bernstein polynomials, actually). In the CAD industry (which is where NURBS are used most, I guess), you can't just make up your own new types of curves, with new types of basis functions. People who use CAD systems often have to exchange data with other people who use other CAD systems, and NURBS are the standard exchange medium -- every system uses them. The benefits of standardisation and easy data exchange far out-weigh any benefits that could be gained by using some exotic new form of curve.

So, people who develop CAD software focus on new/better ways to construct NURBS curves and surfaces, not on making up new types of geometry.

Answer 2

The following four terms are often conflated in the spline curve literature and in many peoples' minds:

• (1) lower quality / higher quality (an ambiguous term, maybe best avoided)
• (2) more local control / more global control
• (3) less smooth / more smooth (an ambiguous term, maybe best avoided)
• (4) number of continuous derivatives

This conflation might be, at least in part, due to the fact that, for the particular cases of B-Spline and NURBS curves, (2) and (4), and at least one definition of (3) and (1), happen to be the same; that is, as we move to higher degree polynomials, we simultaneously get more global control, more continuous derivatives (i.e. "smoother", at least for one definition of the word) and higher quality (again, for at least one definition of the word). Another contributer to the conflation is that the words "smoother" and "smooth" can mean, informally, "higher quality", or it can mean, formally, "has a relatively high number of continuous derivatives" and "$C^{\infty}$" respectively. This all ends up being quite misleading and confusing.

So I'll avoid talking about (1) since the words "high quality" are really meaningless by themselves; really any criterion at all could be considered high quality, depending on the application one has in mind, so I think anyone who uses those words ought to first define them and explain their rationale. I won't attempt to do so here.

Likewise, I'll avoid the words "smoother" and "smooth" since the misleading ambiguity of these words seems to be a big contributor to all of these concepts getting tangled up, as I already said.

So that leaves (2) and (4), which definitely aren't the same. To show they aren't the same, I'll just give an example of a scheme that yields $C^{\infty}$ curves and yet has local control; I think this is pretty much what you had in mind. I don't claim this result is "high quality" in any sense, and I probably wouldn't use it for anything, although I can't put my finger on exactly what about it seems bad.

My example has the same definition as the B-spline basis functions for NURBS curves, except that $f_{i,1}$ and $g_{i,1}$, instead of rising/falling linearly from $0$ to $1$, follow a $C^{\infty}$ transition function from $0$ to $1$ or from $1$ to $0$ respectively.

For comparison, here are the standard NURBS basis functions of various degrees with nine knots $0,1,2,3,4,4.1,5.1,6.1,7.1$, as in the wikipedia article:

And here are the modified NURBS basis functions, constructed with modified $f_{i,1}$ and $g_{i,1}$ as described above: (There is an interactive version of the above pictures, where you can add, move, and delete knots, here on codepen.)

These modified basis functions satisfy the following properties:

• non-negative
• compact support
• partition of unity (true for order 2 since it's true for the modified $f_{i,1}$ and $g_{i,1}$, and can be proved by induction for higher orders, just like for NURBS)
• for order >= 2, the basis functions, and therefore the resulting curves, are $C^{\infty}$ everywhere (unlike NURBS)

Your question "why aren't schemes such as this used" remains unanswered, but maybe at least the question is clearer now.

It's a very interesting question. I would be interested to know what the advantages and disadvantages are of such modified NURBS (or any other such $C^{\infty}$ variation). Aside from uninteresting issues of non-compatibility with existing software, does a scheme such as this have merit or not? It seems like these modified NURBS have all the usual advantages of NURBS, plus $C^{\infty}$ continuity, which is often touted as a good thing, but at this point I'm not sure why.

There's certainly something not-so-nice about these curves-- they seem rather bumpy and unattractive compared to NURBS, although I can't quantify this statement. Can it be quantified reasonably, in a way that makes it clear that NURBS is objectively better? And is there perhaps a variation that is not as bumpy, and more attractive?

I'd like to understand these things better.