Rational Splines
A rational spline is any function of the
form r(x) = s(x)/w(x),
with both s and w splines
and, in particular, w a scalar-valued spline,
while s often is vector-valued.
Rational splines are attractive since it is possible to describe
various basic geometric shapes, like conic sections, exactly as the
range of a rational spline. For example, a circle can so be described
by a quadratic rational spline with just two pieces.
In this toolbox, there is the additional requirement that both s and w be
of the same form and even of the same order, and with the same knot
or break sequence. This makes it possible to store the rational spline r as
the ordinary spline R whose value at x is
the vector [s(x);w(x)].
Depending on whether the two splines are in B-form or ppform, such
a representation is called here the rBform or the rpform of such a
rational spline.
It is easy to obtain r from R.
For example, if v is the value of R at x,
then v(1:end-1)/v(end) is the value of r at x.
There are corresponding ways to express derivatives of r in
terms of derivatives of R.
 | Multivariate Splines | | The ppform |  |
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