D-NURBS are applicable to the optimal fitting of regular or scattered data [28]. The most general and often most useful case occurs with scattered data, when there are fewer or more data points than unknowns--i.e., when the solution is underdetermined or overdetermined by the data. In this case, D-NURBS can yield ``optimal'' solutions by minimizing the thin-plate under tension deformation energy [35, 33]. The surfaces are optimal in the sense that they provide the smoothest curve or surface (as measured by the deformation energy) which interpolates or approximates the data.
The data point interpolation problem amounts to a linear constraint
problem when the weights 
are fixed, and it is amenable to
the constraint techniques presented in Section
5.2. The optimal approximation problem can
be approached in physical terms, by coupling the D-NURBS to the data
through Hookean spring forces (19). We interpret
in (19) as the data point (generally in
) and 
as the D-NURBS parametric coordinates
associated with the data point (which may be the nearest material
point to the data point). The spring constant c determines the
closeness of fit to the data point.
We present three examples of surface fitting using D-NURBS coupled to data points through spring forces. Fig. 3(a) shows 19 data points sampled from a hemisphere and their interpolation with a quadratic D-NURBS surface with 49 control points. Fig. 3(b) shows 19 data points and the reconstruction of the implied convex/concave surface by a quadratic D-NURBS with 49 control points. The spring forces associated with the data points are applied to the nearest points on the surface. In Fig. 3(c) we reconstruct a wave shape from 25 sample points using springs with fixed attachments to a quadratic D-NURBS surface with 25 control points.
Figure 3: Optimal surface fitting: D-NURBS surfaces fit to sampled data
from (a) a hemisphere, (b) a convex/concave surface, (c) a sinusoidal
surface. (a-c1) D-NURBS patch outline with control points (white) and
data points (red) shown. (a-c2) D-NURBS surface at equilibrium fitted
to scattered data points. Red line segments in (c2) represent springs
with fixed attachment points on surface.
