2-3 Curve Fitting

Curve fitting is an important type of interpolation in many applications of GIS.Curve fitting is divided into two categories;

- exact interpolation : a fitted curve passes through all given points
- approximate interpolation : a fitted curve does not always pass through all given points

Exact Interpolation
There are three methods;

- nearest neighbor : the same value as that of the observation is given within the proximal distance, as shown in Figure 2.7.
- linear interpolation: a piecewise linear function is applied between two adjacent points as shown in Figure 2.8.

- cubic interpolation : a third order polynomial is applied between two adjacent points under the condition that the first and
   second order differentials should be continuous. Such a curve is called "spline" (see Figure 2.9).

y = ax3 + bx2 + cx + d

In case, when the curve is not a single function of x as whown in Figure 2.10, an auxilliary variable u should be introduced as follows.

Approximate Interpolation
There are three methods;

- Moving Average : a window with a range of -d to +d is set to average the observation within the region as shown in
   Figure 2.11

- B spline : a cubic curve is determined by using four adjacent observations as shown in Figure 2.12

- Curve Fitting by Least Square Method : see section 2-5