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Multivariate splines can be obtained from univariate splines by the tensor product construct. For example, a trivariate spline in B-form is given by
$$f\left(x,y,z\right)={\displaystyle \sum _{u=1}^{U}{\displaystyle \sum _{v=1}^{V}{\displaystyle \sum _{w=1}^{W}{B}_{u,k}\left(x\right){B}_{v,l}\left(y\right){B}_{w,m}\left(z\right){a}_{u,v,w}}}}$$
with B_{u,k},B_{v,l},B_{w,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 ℜ^{3}.
A very different bivariate spline is the thin-plate spline. This is a function of the form
$$f\left(x\right)={\displaystyle \sum _{j=1}^{n-3}\Psi \left(x-{c}_{j}\right){a}_{j}+x\left(1\right){a}_{n-2}+x\left(2\right){a}_{n-1}+{a}_{n}}$$
with ψ(x)=|x|^{2}log|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 c_{j} are points in ℜ^{2}.
Thin-plate splines arise as bivariate smoothing splines, meaning a thin-plate spline minimizes
$$p{\displaystyle \sum _{i=1}^{n-3}\left|{y}_{i}-f{c}_{i}\right|+\left(1-p\right){\displaystyle \int \left({\left|{D}_{1}{D}_{1}f\right|}^{2}+2{\left|{D}_{1}{D}_{2}f\right|}^{2}+{\left|{D}_{2}{D}_{2}f\right|}^{2}\right)}}$$
over all sufficiently smooth functions f. Here, the y_{i} are data values given at the data sites c_{i}, p is the smoothing parameter, and D_{j}f denotes the partial derivative of f with respect to x(j). The integral is taken over the entire ℜ^{2}. 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.
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.