## Constructing and Working with B-form Splines

### Construction of B-form

Usually, a spline is constructed from some information, like function values and/or
derivative values, or as the approximate solution of some ordinary differential
equation. But it is also possible to make up a spline from scratch, by providing its
knot sequence and its coefficient sequence to the command `spmak`

.

For example, if you enter

sp = spmak(1:10,3:8);

you supply the uniform knot sequence `1:10`

and the coefficient
sequence `3:8`

. Because there are 10 knots and 6 coefficients, the
order must be 4(= 10 – 6), i.e., you get a cubic spline. The command

fnbrk(sp)

prints out the constituent parts of the B-form of this cubic spline, as follows:

knots(1:n+k) 1 2 3 4 5 6 7 8 9 10 coefficients(d,n) 3 4 5 6 7 8 number n of coefficients 6 order k 4 dimension d of target 1

Further, `fnbrk`

can be used to supply each of these parts
separately.

But the point of the Curve Fitting Toolbox™ spline functionality is that there shouldn't be any need for you to look
up these details. You simply use `sp`

as an argument to commands that
evaluate, differentiate, integrate, convert, or plot the spline whose description is
contained in `sp`

.

### Working With B-form Splines

The following commands are available for spline work. There is `spmak`

and `fnbrk`

to make up a spline and take it
apart again. Use `fn2fm`

to convert from B-form to ppform.
You can evaluate, differentiate, integrate, minimize, find zeros of, plot, refine, or
selectively extrapolate a spline with the aid of `fnval`

,
`fnder`

, `fndir`

, `fnint`

,
`fnmin`

, `fnzeros`

, `fnplt`

,
`fnrfn`

, and `fnxtr`

.

There are five commands for generating knot sequences:

`augknt`

for providing boundary knots and also controlling the multiplicity of interior knots`brk2knt`

for supplying a knot sequence with specified multiplicities`aptknt`

for providing a knot sequence for a spline space of given order that is suitable for interpolation at given data sites`optknt`

for providing an*optimal*knot sequence for interpolation at given sites`newknt`

for a knot sequence perhaps more suitable for the function to be approximated

In addition, there is:

To display a spline *curve* with given two-dimensional coefficient
sequence and a uniform knot sequence, use `spcrv`

.

You can also write your own spline construction commands, in which case you will need
to know the following. The construction of a spline satisfying some interpolation or
approximation conditions usually requires a *collocation matrix*, i.e., the matrix that, in each row,
contains the sequence of numbers
*D*^{r}*B*_{j,k}(τ),
i.e., the *r*th derivative at τ of the *j*th B-spline,
for all *j*, for some *r* and some site τ. Such a
matrix is provided by `spcol`

. An optional argument allows for this matrix to be
supplied by `spcol`

in a space-saving spline-almost-block-diagonal-form
or as a MATLAB^{®} sparse matrix. It can be fed to `slvblk`

, a command for
solving linear systems with an almost-block-diagonal coefficient matrix. If you are
interested in seeing how `spcol`

and `slvblk`

are used
in this toolbox, have a look at the commands `spapi`

,
`spap2`

, and `spaps`

.

In addition, there are routines for constructing *cubic* splines.
`csapi`

and `csape`

provide the cubic spline
interpolant at knots to given data, using the not-a-knot and various other end
conditions, respectively. A parametric cubic spline curve through given points is
provided by `cscvn`

. The cubic *smoothing* spline is
constructed in `csaps`

.

### Example: B-form Spline Approximation to a Circle

As another simple example,

points = .95*[0 -1 0 1;1 0 -1 0]; sp = spmak(-4:8,[points points]);

provides a planar, quartic, spline curve whose middle part is a pretty good approximation to a circle, as the plot on the next page shows. It is generated by a subsequent

plot(points(1,:),points(2,:),'x'), hold on fnplt(sp,[0,4]), axis equal square, hold off

Insertion of additional control points $$\left(\pm 0.95,\pm 0.95\right)/\sqrt{1.9}$$ would make a visually perfect circle.

Here are more details. The spline curve
generated has the form Σ^{8}_{j=1}B_{j,5}*a*(:,* j*),
with -`4:8 `

the uniform knot sequence, and with its
control points *a*(:,*j*) the sequence (0,α),(–α,0),(0,–α),(α,0),(0,α),(–α,0),(0,–α),(α,0) with α=0.95.
Only the curve part between the parameter values 0 and 4 is actually
plotted.

To get a feeling for how close to circular this part of the
curve actually is, compute its unsigned curvature. The curvature κ(*t*)
at the curve point γ(*t*) = (x(*t*),
y(*t*)) of a space curve γ can be computed
from the formula

$$\kappa =\frac{\left|x\text{'}y\text{'}\text{'}-y\text{'}x\text{'}\text{'}\right|}{{(x{\text{'}}^{2}+y{\text{'}}^{2})}^{3/2}}$$

in which x', x″, y', and y” are the first and
second derivatives of the curve with respect to the parameter used
(*t*). Treat the planar curve as a space curve in
the (*x*,*y*)-plane, hence obtain
the maximum and minimum of its curvature at 21 points as follows:

t = linspace(0,4,21);zt = zeros(size(t)); dsp = fnder(sp); dspt = fnval(dsp,t); ddspt = fnval(fnder(dsp),t); kappa = abs(dspt(1,:).*ddspt(2,:)-dspt(2,:).*ddspt(1,:))./... (sum(dspt.^2)).^(3/2); [min(kappa),max(kappa)] ans = 1.6747 1.8611

So, while the curvature is not quite constant, it is close to 1/radius of the circle, as you see from the next calculation:

1/norm(fnval(sp,0)) ans = 1.7864

**Spline Approximation to a Circle; Control Points Are Marked
x**