22. 
The methods of the first three groups which are discussed have been 
designed for use with a random pattern and an arbitrary number of reference points. 
These methods can require a considerable amount of computation. They are best 
used to compute or are especially designed to compute, in first instance, heights 
at the nodes of a regular grid and are then supplemented by a method for inter- 
polation between the nodes. These methods can perform smoothing of the data as 
well as interpolation. The methods of the remaining three groups perform separate 
local interpolations between the nearest reference points. 
The methods of the first group compute for each interpolated point the 
surface whose weighted sum of squares of distances to the reference points is a 
minimum. The height of the interpolated point is here found as the height of the 
computed surface at that point. The principal differences between the methods of 
this group consist in the type of surface and the weight function that are used. 
The methods of the second group construct around each reference point a 
fixed surface which has a vertical axis of symmetry at that point. In a specific 
method, these surfaces differ only in vertical scale. In different methods, their 
profiles are differently shaped. Interpolation is performed by summing the heights 
of all surfaces. 
The methods of the third group first cover the interpolation region with 
a regular grid which is independent of the position of the reference points. They 
then construct in each element of this grid a local polynomial surface. The heights 
and possibly other parameters at all the nodes of the grid are computed simulta- 
neously, considering a best possible fit of the local surfaces at the reference 
points and continuity and possibly smoothness of the total surface. 
The fourth group interpolates between reference points located at the 
nodes of a rectangular grid. As in the case of the third group, in each grid 
element a local polynomial surface is constructed. 
The fifth group interpolates in a net of triangles whose vertices are at 
the reference points. Included here are methods based upon a random reference 
point pattern and methods based upon a regular pattern. In most, but not in all 
of these methods, the interpolation in the triangles is linear. 
The sixth and last group includes all methods which interpolate in areas 
surrounded by reference points located on characteristic terrain lines. 
Another group could be formed from methods which construct one 
analytical function to represent the terrain surface in the whole region of 
interest. Such a function is discussed occasionally, but it is almost generally 
found to be too little adaptable to the terrain surface. The least unsuitable 
appears to be the use of orthogonal polynomials [70]. 
3. Moving surface methods 
Under this heading, all methods can be collected which require for each 
interpolated point the computation of a surface. Generally, going from one 
interpolated point to an adjacent one, this surface will change its orientation 
and, possibly, its shape. For this reason, it has been called a moving or roving 
surface. 
The height of an interpolated point is found as the height of the 
instantaneous position of the surface at that point. The surface will be either 
a level plane, a tilting plane, or a second-degree surface. It is defined 
algebraically by one or more terms of the equation 
h=aq +ayx+ ay + bıx? + boxy + bay? (1)