## Multivariate and Rational Splines

### Multivariate Splines

Multivariate splines can be obtained from univariate splines by the tensor product construct. For example, a trivariate spline in B-form is given by

`$f\left(x,y,z\right)=\sum _{u=1}^{U}\sum _{v=1}^{V}\sum _{w=1}^{W}{B}_{u,k}\left(x\right){B}_{v,l}\left(y\right){B}_{w,m}\left(z\right){a}_{u,v,w}$`

with Bu,k,Bv,l,Bw,m univariate B-splines. Correspondingly, this spline is of order k in x, of order l in y, and of order m in z. Similarly, the ppform of a tensor-product spline is specified by break sequences in each of the variables and, for each hyper-rectangle thereby specified, a coefficient array. Further, as in the univariate case, the coefficients may be vectors, typically 2-vectors or 3-vectors, making it possible to represent, e.g., certain surfaces in ℜ3.

A very different bivariate spline is the thin-plate spline. This is a function of the form

`$f\left(x\right)=\sum _{j=1}^{n-3}\Psi \left(x-{c}_{j}\right){a}_{j}+x\left(1\right){a}_{n-2}+x\left(2\right){a}_{n-1}+{a}_{n}$`

with ψ(x)=|x|2log|x|2 the thin-plate spline basis function, and |x| denoting the Euclidean length of the vector x. Here, for convenience, denote the independent variable by x, but x is now a vector whose two components, x(1) and x(2), play the role of the two independent variables earlier denoted x and y. Correspondingly, the sites cj are points in ℜ2.

Thin-plate splines arise as bivariate smoothing splines, meaning a thin-plate spline minimizes

`$p\sum _{i=1}^{n-3}|{y}_{i}-f{c}_{i}{}^{2}|+\left(1-p\right)\int \left({|{D}_{1}{D}_{1}f|}^{2}+2{|{D}_{1}{D}_{2}f|}^{2}+{|{D}_{2}{D}_{2}f|}^{2}\right)$`

over all sufficiently smooth functions f. Here, the yi are data values given at the data sites ci, p is the smoothing parameter, and Djf denotes the partial derivative of f with respect to x(j). The integral is taken over the entire ℜ2. The upper summation limit, n–3, reflects the fact that 3 degrees of freedom of the thin-plate spline are associated with its polynomial part.

Thin-plate splines are functions in stform, meaning that, up to certain polynomial terms, they are a weighted sum of arbitrary or scattered translates Ψ(· -c) of one fixed function, Ψ. This so-called basis function for the thin-plate spline is special in that it is radially symmetric, meaning that Ψ(x) only depends on the Euclidean length, |x|, of x. For that reason, thin-plate splines are also known as RBFs or radial basis functions. See Constructing and Working with stform Splines for more information.

### 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 because 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.