Methods for scattered data interpolation and fitting are often used for surface design. Usually, when designing a surface, due to the physical restriction of the data measurements, an interpolatory method is often preferrable. A set of given data (x_i, y_i, z_i)'s in 3D is shown in the left of the following figures. They are data for a car panel. Image that one has to replace a damaged panel of an antique car and the part could not be found. The data set is a set of the measurements of the panel. Let us regenerate the surface of the car panel on computer and then rebuild it. We can use bivariate splines to do that. In the right of the following figures, a C^1 quintic spline interpolatory surface is generated using the minimal energy method.

In addition to the minimal energy method, there are many other useful interpolatory and fitting methods for surface design. Also it may be necessary to mix interpolation and fitting techniques together. For example, given a set of scattered data which is assumed to be sampled from a smooth convex function or assumed to have a piecewise linear interpolant which is convex, a practical and theoretical problem is to find a convex C^1 spline interpolant. Similarly, for a given monotonic data or positive data, practical and theoretical problems are to find a monotonic or positive C^1 or higher order smooth spline interpolants. These problems in the bivariate setting, called shape preservation interpolation problems are still open so far.

To make the problem trackable, we have to reduce the requirement of interpolation when constructing shape preserving smooth spline surfaces. That is, divide the data set into two classes: one is the data values are absolutely fixed and must be interpolated and the other is the remaining part of the data. For example, when designing the roof or the hood of a car body, the data on the four sides of the roof or hood have no negotiation, i.e., the surface has to meet these values in order to to be well assembled with other parts of the car body. However, the top part of the roof or the central part of hood have a little bit flexibility. Hence, one may combine interpolation and fitting techniques together by minimizing the energy functional and least squares of differences between the spline values and given data values (only over the central part of hood) subject to the interpolatory conditions only along the four sides (the boundary of the hood). This method may be called a mixed fitting method. For another example, when the given data admits a convex linear interpolant, one may sample many data values from the convex linear interpolant and use them to find the smooth interpolation/fitting spline by the mixed fitting method above. As the triangulation size gets small and the amount of data values from the convex linear interpolant increases, the smooth spline surface is closed to the convex interpolant and hence approximately satisfies the interpolation and convex constraints. Although this proposed method does not solve the shape preservation problem, the constraints of interpolation and shape preservation are weakened so that a smooth surface with reasonable interpolation and shape preservation can be achieved.

In the above figures, a set of data is obtained by measuring the hood of a truck model (1997 Ford F150 Pickup) manually. They are not symmetric due to the human measurement error. A type-I triangulation can be used in this case. Using the standard minimal energy method, the contours show that these surfaces are not convex and fairing enough. Next by swapping triangles, one finds a linear interpolant which has the minimal surface area. Then using the data values sampling from the linear interpolant of minimal surface area and its associate triangulation, we apply the mixed fitting method mentioned above to find an interpolation/fitting C^1 quintic spline surface. The contours show that it is much nicer than those by the minimal energy C^1 quintic spline. More examples can be found in the webpage by Ms. Dustin Burns.

Bivariate splines can be also used to fill some holes in a surface. A webpage containing some artifical examples can be found here These examples are generated by using C^1 quintic splines based on the minimization of the energy functional subject to the Hermite interpolation (function values and normal derivatives) on the boundary of the hole.