### Introduction to Rational Splines

A rational
spline is, by definition, any function that is the ratio of two splines:

$$r\left(x\right)=s\left(x\right)/w\left(x\right)$$

This requires *w* to be scalar-valued, but *s* is
often chosen to be vector-valued. Further, it is desirable that *w*(*x*)
be not zero for any *x* of interest.

Rational splines are popular because, in contrast to ordinary
splines, they can be used to describe certain basic design shapes,
like conic sections, exactly.

### rsform: rpform, rBform

The two splines, *s* and *w*,
in the rational spline *r*(*x*)=*s*(*x*)/*w*(*x*)
need not be related to one another. They could even be of different
forms. But, in the context of this toolbox, it is convenient to restrict
them to be of the same form, and even of the same order and with the
same breaks or knots. For, under that assumption, you can represent
such a rational spline by the (vector-valued) spline function

$$R\left(x\right)=\left[s\left(x\right);w\left(x\right)\right]$$

whose values are vectors with one more entry than the values
of the rational spline *r*, and call this the *rsform* of the rational
spline, or, more precisely, the *rpform* or *rBform*, depending on whether *s* and *w* are
in ppform or in B-form. Internally, the only thing that distinguishes
these rational forms from their corresponding ordinary spline forms,
rpform and B-form, is their form part, i.e., the string obtained
via `fnbrk(r,'form')`

. This is enough to alert the `fn...`

commands
to act appropriately on a function in one of the rsforms.

For example, as is done in `fnval`

, it is very
easy to obtain *r*(*x*) from *R*(*x*).
If `v`

is the value of *R* at x,
then `v(1:end-1)/v(end)`

is the value of *r* at *x*.
If, in addition, `dv`

is *DR*(*x*),
then `(dv(1:end-1)-dv(end)*v(1:end-1))/v(end)`

is *Dr*(*x*).
More generally, by Leibniz's formula,

$${D}^{j}s={D}^{j}\left(wr\right)={\displaystyle \sum _{i=0}^{j}\left(\begin{array}{c}j\\ i\end{array}\right)}{D}^{i}w{D}^{j-i}r$$

Therefore,

$${D}^{j}r=\left({D}^{j}s-{\displaystyle \sum _{i=1}^{j}\left(\begin{array}{c}j\\ i\end{array}\right){D}^{i}w{D}^{j-i}r}\right)/w$$

This shows that you can compute the derivatives of *r* inductively,
using the derivatives of *s* and *w* (i.e.,
the derivatives of *R*) along with the derivatives
of *r* of order less than *j* to
compute the *j*th derivative of *r*.
This inductive scheme is used in `fntlr`

to provide
the first so many derivatives of a rational spline. There is a corresponding
formula for partial and directional derivatives for multivariate rational
splines.