## The B-form

### Introduction to B-form

A univariate spline *f* is specified by its nondecreasing **
**knot sequence `t`

and by its B-spline coefficient sequence
`a`

. See Multivariate Tensor Product Splines for a discussion
of multivariate splines. The coefficients may be (column-)vectors, matrices, even
ND-arrays. When the coefficients are 2-vectors or 3-vectors, *f* is a curve in
R^{2} or R^{3} and the coefficients are
called the *
**control points* for the curve.

Roughly speaking, such a spline is a piecewise-polynomial of a certain order and with
breaks `t`

(*i*). But knots are different from breaks in that they may be repeated, i.e., `t`

need not be
*strictly* increasing. The resulting knot
*multiplicities* govern the smoothness of the spline across the knots, as detailed below.

With `[d,n] = size(a)`

, and` n+k = length(t)`

, the
spline is of *order*
`k`

. This means that its polynomial pieces have degree <```
k
```

. For example, a *cubic* spline is a spline of *order 4*
because it takes four coefficients to specify a cubic polynomial.

### Definition of B-form

These four items, *t*, *a*, *n*,
and *k*, make up the B-form of the spline *f*.

This means, explicitly, that

$$f={\displaystyle \sum _{i=1}^{n}{B}_{i,k}a\left(:,i\right)}$$

with *B _{i,k}=B(·|t(i:i+k))* the

*i*th B-spline of order

`k`

for
the given knot sequence `t`

, i.e., the B-spline with knots
*t(i),...,t(i+k)*. The basic interval of this B-form is the interval

*[t(1)..t(n+k)]*. It is the default interval over which a spline in B-form is plotted by the command

`fnplt`

. Note that a spline in B-form
is zero outside its basic interval while, after conversion to ppform via
`fn2fm`

, this is usually not the case because, outside its basic
interval, a piecewise-polynomial is defined by extension of its first or last polynomial piece. In particular, a function
in B-form may have jumps in value and/or one of its derivative not only across its interior knots, i.e., across *t*(

*i*) with

*t(1)<t(i)<t(n+k)*, but also across its end knots,

*t*(1) and

*t*(

*n*+

*k*).

### B-form and B-Splines

The building blocks for the B-form of a spline are the B-splines. A B-Spline of Order 4, and the Four Cubic Polynomials from Which It Is Made shows a picture of such a B-spline, the one
with the knot sequence `[0 1.5 2.3 4 5]`

, hence of order 4, together
with the polynomials whose pieces make up the B-spline. The information for that picture
could be generated by the command

To summarize: The B-spline with knots t(*i*)≤····≤
t(*i*+*k*) is positive on the interval (t(*i*)..t(*i*+*k*))and is zero outside that interval. It is piecewise-polynomial of order
`k`

with breaks at the sites t(*i*),...,t(*i*+*k*). These knots may coincide, and the precise
*multiplicity* governs the smoothness with which the two
polynomial pieces join there.

#### Definition of B-Splines

The shorthand

$$f\in {S}_{k,t}$$

is one of several ways to indicate that *f* is a spline of order
`k`

with knot sequence `t`

, i.e., *a
linear combination of the B-splines* of order `k`

for
the knot sequence `t`

.

A word of caution: The term *B-spline* has been expropriated by the Computer-Aided
Geometric Design (CAGD) community to mean what is called here a *spline in
B-form*, with the unhappy result that, in any discussion between
mathematicians/approximation theorists and people in CAGD, one now always has to
check in what sense the term is being used.

### B-Spline Knot Multiplicity

The rule is

**knot multiplicity + condition multiplicity = order**

**All Third-Order B-Splines for a Certain Knot Sequence with
Various Knot Multiplicities**

For example, for a B-spline of order 3, a simple knot would mean two smoothness conditions, i.e., continuity of function and first derivative, while a double knot would only leave one smoothness condition, i.e., just continuity, and a triple knot would leave no smoothness condition, i.e., even the function would be discontinuous.

All Third-Order B-Splines for a Certain Knot Sequence with Various Knot Multiplicities shows a picture of all the third-order B-splines for a certain mystery knot sequence
`t`

. The breaks are indicated by vertical lines. For each break, try
to determine its multiplicity in the knot sequence (it is 1,2,1,1,3), as well as its
multiplicity as a knot in each of the B-splines. For example, the second break has
multiplicity 2 but appears only with multiplicity 1 in the third B-spline and not at
all, i.e., with multiplicity 0, in the last two B-splines. Note that only one of the
B-splines shown has all its knots simple. It is the only one having three different
nontrivial polynomial pieces. Note also that you can tell the knot-sequence multiplicity
of a knot by the number of B-splines whose nonzero part begins or ends there. The
picture is generated by the following MATLAB^{®} statements, which use the command `spcol`

from this toolbox to generate the function values of all
these B-splines at a fine net `x`

.

t=[0,1,1,3,4,6,6,6]; x=linspace(-1,7,81); c=spcol(t,3,x);[l,m]=size(c); c=c+ones(l,1)*[0:m-1]; axis([-1 7 0 m]); hold on for tt=t, plot([tt tt],[0 m],'-'), end plot(x,c,'linew',2), hold off, axis off

Further illustrated examples are provided by the example ”Construct and Work
with the B-form”. You can also use the GUI `bspligui`

to study
the dependence of a B-spline on its knots experimentally.

### Choice of Knots for B-form

The rule “knot multiplicity + condition multiplicity = order” has the
following consequence for the process of choosing a knot sequence for the B-form of a
spline approximant. Suppose the spline *s* is to be of order
*k*, with basic interval
[*a*..*b*], and with interior breaks ξ_{2}< ··
·<ξ_{l}. Suppose, further, that, at
ξ_{i}, the spline is to satisfy μ_{i}
smoothness conditions, i.e.,

$$\begin{array}{ccc}jum{p}_{{\xi}_{i}}{D}^{j}s:={D}^{j}s\left({\xi}_{i+}\right)-{D}^{j}s\left({\xi}_{i-}\right)=0,& 0\le j<{\mu}_{i},& i=2,\mathrm{...},l\end{array}$$

Then, the appropriate knot sequence *t* should contain the break ξ_{i} exactly *k* –
μ_{i} times, *i*=2,...,*l*. In addition, it should contain the two endpoints,
*a* and *b*, of the basic interval exactly
*k* times. This last requirement can be relaxed, but has become
standard. With this choice, there is exactly one way to write each spline
*s* with the properties described as a weighted sum of the B-splines of order
*k* with knots a segment of the knot sequence *t*.
This is the reason for the *B* in *B-spline*:
B-splines are, in Schoenberg's terminology, *basic* splines.

For example, if you want to generate the B-form of a cubic spline on the interval [1 .. 3], with interior breaks 1.5, 1.8, 2.6, and with two continuous derivatives, then the following would be the appropriate knot sequence:

t = [1, 1, 1, 1, 1.5, 1.8, 2.6, 3, 3, 3, 3];

This is supplied by `augknt([1, 1.5, 1.8, 2.6, 3], 4)`

. If you
wanted, instead, to allow for a corner at 1.8, i.e., a possible jump in the first derivative there, you would triple the knot 1.8, i.e.,
use

t = [1, 1, 1, 1, 1.5, 1.8, 1.8, 1.8, 2.6, 3, 3, 3, 3];

and this is provided by the statement

t = augknt([1, 1.5, 1.8, 2.6, 3], 4, [1, 3, 1] );