Curve and Surface Fitting

Abstract

In Chapters 7 and 8 we showed how to construct NURBS representations of common and relatively simple curves and surfaces such as circles, conics, cylinders, surfaces of revolution, etc. These entities can be specified with only a few data items, e.g., center point, height, radius, axis of revolution, etc. Moreover, the few data items uniquely specify the geometric entity. In this chapter we enter the realm of free-form (or sculptured) curves and surfaces. We study fitting,i.e., the construction of NURBS curves and surfaces which fit a rather arbitrary set of geometric data, such as points and derivative vectors. We distinguish two types of fitting, interpolation and approximation. In interpolation we construct a curve or surface which satisfies the given data precisely, e.g., the curve passes through the given points and assumes the given derivatives at the prescribed points. Figure 9.1 shows a curve interpolating five points and the first derivative vectors at the endpoints. In approximation, we construct curves and surfaces which do not necessarily satisfy the given data precisely, but only approximately. In some applications --- such as generation of point data by use of coordinate measuring devices or digitizing tablets, or the computation of surface/surface intersection points by marching methods --- a large number of points can be generated, and they can contain measurement or computational noise. In this case it is important for the curve or surface to capture the ``shape'' of the data, but not to ``wiggle'' its way through every point.