7. Concluding remarks

where pnð Þx is a linear polynomial in two or three dimensions:

20 Topics in Splines and Applications

p2ð Þ¼ x ax þ by þ c

polynomials is enforced, for instance, in two dimensions this yields to X j

> <sup>s</sup> xi � � <sup>¼</sup> <sup>X</sup> j

interpolated. The resultant linear system to solve for is of ð Þ m þ n þ 1 rank.

By considering this result, the interpolation problem becomes

Figure 21. Discrete MC data with Duchon splines.

<sup>λ</sup><sup>j</sup> <sup>¼</sup> <sup>X</sup> j

Notice that λ<sup>j</sup> and the polynomial coefficients are all scalar quantities. In order to guarantee existence and uniqueness for these splines, an orthogonality condition with respect to linear

> <sup>λ</sup>jxj <sup>¼</sup> <sup>X</sup> j

<sup>λ</sup><sup>j</sup> � <sup>φ</sup> <sup>r</sup><sup>i</sup> j � �

which implies <sup>m</sup> points plus <sup>n</sup> <sup>þ</sup> 1 orthogonality conditions; here, Fi are the nodal values to be

Duchon splines are certainly suitable to interpolate scattered data sets that we cannot tackle with the tensor product surfaces that we discussed before. Indeed, Figure 21 depicts such an application, in optimization, where an objective function that we wish to minimize was sampled randomly by Monte-Carlo (MC) realizations. To compute a minimum, we interpolate the black dots, and then we minimize the resulting spline with standard Newton stochastic techniques [15]. It is true that Duchon splines are a valid choice for "surrogate" models for such applications.

<sup>p</sup>3ð Þ¼ <sup>x</sup> dx <sup>þ</sup> ey <sup>þ</sup> fz <sup>þ</sup> g ; <sup>λ</sup>j, a, …, g<sup>∈</sup> <sup>R</sup>: (37)

<sup>þ</sup> pn xi � � <sup>¼</sup> Fi

λjyj ¼ 0: (38)

, (39)

We presented a concise introduction to scalar and parametric spline interpolants. We introduced cubic and tension splines for scalar functions, and then we generalized them for the parametric case via Bèzier, B-spline, and NURBS curves. These latter entities are of the particular interest for applications in CAGD. We thus elaborated on topics such as inverse design and interpolation. We extended the treatment also to cover interpolation and translational surfaces with examples in mechanical and petroleum engineering. We wrapped up with the topic of interpolating sparse very high dimensional data sets via Duchon splines which are a kind of response surfaces suitable for applications in statistical analysis and optimization.
