Curve Fitting and Curve Displaying

In Section 7.6 we discussed how one can define a curve on a discrete grid, and in Section 9.7 we discussed how to derive curves from skeletons. The representation of a curve as a sequence of pixels may be adequate for some applications, but for others we would like to have a mathematical expression for it. Such an expression may be far more compact than the discrete forms, as a comparison between Figure 9.8 and Table 9.1 shows. Finding a curve that passes through a set of given points is the problem of interpolation, while finding a curve that passes near a set of given points is the problem of approximation. We shall use the term curve fitting to refer collectively to both of them. The problem of curve displaying is also of interest: given a mathematical expression, identify the pixels that must be marked so that an image of the curve described by the expression is displayed. The problem is by no means trivial, even in the case of straight line displays (Section 7.6).