Approximation of Curves

Abstract

The last two chapters dealt mostly with interpolation where a curve must pass through all the data points. (Guiding points are part of the design parameters, so that a curve defined by them can still be used for interpolation.) There are many applications where interpolation is not necessary, or even desirable, and one wants the curve to pass only near the data points. This is the approximation problem. If the curve fitting is done in an interactive way the distinction between the two problems is not essential. The user modifies parameters (such as guiding points) until the curve looks right. “Looking right” may mean that the curve passes through all the data points, or through most of them, or near all of them, and so on. If the curve fitting is done automatically, such subjective criteria must be replaced by mathematically precise measures of closeness. The most common such measures are the maximum error and the integral square error (ISE). The error can be measured either along a coordinate or along a normal to the approximating curve. The latter is intuitively more appealing but more difficult to compute. Let e _i denote the pointwise error at the i ^th point, i.e. the distance between the curve and the point (measured by either of the above techniques).