- 14 - 
X: gy 2 
zi yi 2) 
€2 y2 95 
X3 Y3 23 
(14) 
In the case of a regular grid, this formula becomes very simple if one coordinate 
axis is taken parallel to one of the sets of parallel lines and the origin is 
shifted to one of the reference points. 
This formulation is used in the Finnish DTM, described by Viita [1-4], 
[2-2] both for irregular and for regular triangles. An exception is here made 
where, in the first case, an interpolated point lies outside the triangle formed 
by the three nearest reference points. A double linear interpolation is then 
performed with the four nearest points. This use of two different interpolation 
procedures cannot produce a continuous surface. 
Beyer [21] performs first a linear interpolation, using the two nearest 
reference points and a suitably located third point. He avoids a discontinuous 
representation by using this interpolation only for the vertices of a net of equi- 
lateral triangles. Subsequent to this, a linear interpolation is performed in 
that net. 
Düppe and Gottschalk [26] have avoided a discontinuous representation by 
devising an algorithm that first subdivides the whole interpolation region into 
optimum triangles. Each interpolated point is then interpolated in the optimum 
triangle inside which it is located. Berger [20] describes a program, called 
TRASS-OPTI, in which use is made of a regular grid only and, therefore, the dis- 
continuity problem does not occur. 
In the case of a regular net of triangles, just as in the case of a 
rectangular grid, a more sophisticated interpolation method would allow a wider 
spacing of the grid points. An obvious possibility would be to interpolate the 
height at the middle of each triangle side by means of eqs. (12a) or (12b), using 
the four nearest points on the grid line on which the point is located. Each 
triangle could then be subdivided into four equilateral triangles, in each of 
which a linear interpolation would be performed. 
A much more sophisticated interpolation method, suitable even for 
complicated non-topographical surfaces and an irregular pattern of reference points, 
is described by Bauhuber et al [10-4]. The interpolation region is divided into 
triangles with vertices at the reference points, following which a continuous and 
smooth surface is constructed as follows. At each reference point a plane is 
constructed which will be tangent to the surface. The sum of squares of distances 
from each plane to the directly connected surrounding reference points is made a 
minimum. Each profile along a triangle side is a cubic polynomial. The cross 
tilt of a tangent plane at a point on a side is linearly interpolated between the 
cross tilts at the ends of the side. Interpolation of the height of a point 
inside a triangle is also by means of cubic polynomials, separately on each of 
three lines parallel to the three sides. Smoothness of the surface is obtained by 
taking the weighted mean of the three computed heights, using a weight function 
that approaches zero when a parallel line approaches a side. 
8. Interpolation in a string DTM 
  
Several of the earlier DTMs are based upon the use of reference points 
located on characteristic terrain lines. Interpolation is here performed in 
profiles and is linear between intersections of the profile with terrain lines. 
Linkwitz [54] describes a DTM which makes use of contour lines. Contour lines and 
break lines are used also in the Highway Optimization Program System HOPS of the 
British Transport and Road Research Laboratory and in the DTM developed by 
Northamptonshire County Council [6], [5].