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For example,

circle = rsmak('circle');

provides a rational spline whose values on its basic interval trace out the unit circle, i.e., the circle of radius 1 with center at the origin, as the command

fnplt(circle), axis square

readily shows; the resulting output is the circle in the figure A Circle and an Ellipse, Both Given by a Rational Spline.

It is easy to manipulate this circle to obtain related shapes. For example, the next commands stretch the circle into an ellipse, rotate the ellipse 45 degrees, and translate it by (1,1), and then plot it on top of the circle.

ellipse = fncmb(circle,[2 0;0 1]); s45 = 1/sqrt(2); rtellipse = fncmb(fncmb(ellipse, [s45 -s45;s45 s45]), [1;1] ); hold on, fnplt(rtellipse), hold off

As a further example, the "circle" just constructed is put together from four pieces. To highlight the first such piece, use the following commands:

quarter = fnbrk(fn2fm(circle,'rp'),1); hold on, fnplt(quarter,3), hold off

In the first command, `fn2fm`

is used to change
forms, from one based on the B-form to one based on the ppform, and
then `fnbrk`

is used to extract the first piece,
and this piece is then plotted on top of the circle in A Circle and an Ellipse, Both Given by a Rational Spline, with linewidth `3`

to
make it stand out.

**A Circle and an Ellipse, Both Given by a Rational Spline**

As a surface example, the command `rsmak('southcap')`

provides
a 3-vector valued rational bicubic polynomial whose values on the
unit square [-1 .. 1]^2 fill out a piece of the unit sphere. Adjoin to it five suitable rotates of it and you
get the unit sphere exactly. For illustration, the following commands
generate two-thirds of that sphere, as shown in Part of a Sphere Formed by Four Rotates of a Quartic Rational.

southcap = rsmak('southcap'); fnplt(southcap) xpcap = fncmb(southcap,[0 0 -1;0 1 0;1 0 0]); ypcap = fncmb(xpcap,[0 -1 0; 1 0 0; 0 0 1]); northcap = fncmb(southcap,-1); hold on, fnplt(xpcap), fnplt(ypcap), fnplt(northcap) axis equal, shading interp, view(-115,10), axis off, hold off

**Part of a Sphere Formed by Four Rotates of a Quartic Rational**

Having chosen to represent the rational spline *r* = *s*/*w* in
this way by the ordinary spline *R*=[*s*;*w*] makes
it is easy to apply to a rational spline all the `fn...`

commands
in the Curve Fitting Toolbox™ spline functions, with the following
exceptions. The integral of a rational spline need not be a rational
spline, hence there is no way to extend `fnint`

to
rational splines. The derivative of a rational spline *is* again
a rational spline but one of roughly twice the order. For that reason, `fnder`

and `fndir`

will
not touch rational splines. Instead, there is the command `fntlr`

for
computing the value at a given `x`

of all derivatives
up to a given order of a given function. If that function is rational,
the needed calculation is based on the considerations given in the
preceding paragraph.

The command `r = rsmak(shape)`

provides rational
splines in rBform that describe exactly certain standard geometric
shapes , like `'circle'`

, `'arc'`

, `'cylinder'`

, `'sphere'`

, `'cone'`

, `'torus'`

.
The command `fncmb(r,trans)`

can be used to apply
standard transformations to the resulting shape. For example, if `trans`

is
a column-vector of the right length, the shape would be translated
by that vector while, if `trans`

is a suitable matrix
like a rotation, the shape would be transformed by that matrix.

The command `r = rscvn(p)`

constructs the quadratic
rBform of a tangent-continuous curve made up of circular arcs and
passing through the given sequence, `p`

, of points
in the plane.

A special rational spline form, called a NURBS, has become a standard tool in CAGD. A NURBS is, by definition, any rational spline for
which both *s* and *w* are in the
same B-form, with each coefficient for *s* containing
explicitly the corresponding coefficient for *w* as
a factor:

$$\begin{array}{cc}s={\displaystyle \sum _{i}{B}_{i}v\left(i\right)a\left(:,i\right),}& w={\displaystyle \sum _{i}{B}_{i}v\left(i\right)}\end{array}$$

The normalized coefficients *a*(:,*i*)
for the numerator spline are more readily used as control points than
the unnormalized coefficients *v*(*i*)*a*(:,*i*)
used in the rBform. Nevertheless, this toolbox provides no special
NURBS form, but only the more general rational spline, but in both
B-form (called `rBform`

internally) and in ppform
(called `rpform`

internally).

The rational spline `circle`

used earlier is
put together in `rsmak`

by code like the following.

x = [1 1 0 -1 -1 -1 0 1 1]; y = [0 1 1 1 0 -1 -1 -1 0]; s45 = 1/sqrt(2); w =[1 s45 1 s45 1 s45 1 s45 1]; circle = rsmak(augknt(0:4,3,2), [w.*x;w.*y;w]);

Note the appearance of the denominator spline as the last component.
Also note how the coefficients of the denominator spline appear here
explicitly as factors of the corresponding coefficients of the numerator
spline. The normalized coefficient sequence `[x;y]`

is
very simple; it consists of the vertices and midpoints, in proper
order, of the "unit square". The resulting control
polygon is tangent to the circle at the places where the four quadratic
pieces that form the circle abut.

For a thorough discussion of NURBS, see [G. Farin, *NURBS*,
2nd ed., AKPeters Ltd, 1999] or [Les Piegl and Wayne Tiller, *The
NURBS Book*, 2nd ed., Springer-Verlag, 1997].

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