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For the interpolation of a point, the coordinates and the height of each 
of the surrounding reference points are substituted into this equation, each 
reference height is given a weight that is a monotonic decreasing function of the 
distance to the reference point, and the parameters in the equation are solved by 
the method of least squares. Taking the origin of the coordinate system in the 
interpolated point, its height is ag, and only this parameter is computed. 
As a result of the weighting applied to the reference heights, the 
totality of interpolated points defines a representation of the terrain surface 
that cannot be given an analytical formulation except in the case where all 
reference points lie in a plane or on a second-degree surface. 
The crudest and probably the oldest method of this type uses a level 
plane and simply equates the height of this plane with the height of one of the 
nearest reference points. This results in a discontinuous representation of the 
terrain by a set of level surfaces of different heights. Maps constructed in this 
way are called proximal maps and chloropleth maps, see Peucker [7]. Programs 
which can construct such maps are SYMAP of the Laboratory for Computer Graphics 
and Spatial Analysis, Harvard University, and GEOMAP of the University of Waterloo, 
Canada. 
À less crude representation is obtained by making the height of the level 
plane a weighted arithmetic mean of the heights of selected surrounding reference 
points. This method is used in the ECU automated contouring system of the 
Experimental Cartography Unit, Royal College of Art, London, described by 
Connelly [25] and by Rhind [63]. It is used also in SYMAP and in the General 
Purpose Contouring Program GPCP of Calcomp [12]. 
Still better representations can be obtained by using a tilted plane or 
a second-degree surface. The latter can, of course, adapt itself best to the shape 
of the terrain, but it requires more reference points and reportedly can give large 
errors in cases of few reference points. The use of a third-degree polynomial 
increases that danger and does not appreciably improve the results. 
The tilted plane is used by Juncaitis and Junkins [38] in work done for 
the USA TOPOCOM to interpolate heights at the nodes of a square grid. It is used 
also in two German DTMs, one described by Koch [43] and by Fuchs [10-3] and one 
described by Gruber [30]. 
The full second-degree equation has been used since long in France, see 
Baussart [1-1]. It has been selected also for the Numeric Ground Image system 
sponsored by the Texas Highway Department and described by Maxwell [60]. 
Nordin [1-3] records its use, without the xy term, by the Swedish National Road 
Service. 
Koch [41] describes the use of the level plane, the tilted plane, and 
the full second-degree surface as three distinct possibilities. ^ Haendel et al 
[31] describe the program HOELI developed at the University of Dortmund, in which 
the choice between these three surfaces depends upon the number of surrounding 
reference points. Rapior and Bopp [10-5] describe the development of an IBM- 
Germany program in which in general the full second-degree equation is used but in 
which near breaklines and in areas of too sparse control the equation is restricted 
to the linear terms. Sima [67] relates that in the Czechoslovakian DTM the full 
second-degree equation is used in the case of 8 or more reference points. With 
6 or 7 points the xy term is dropped, with 4 or 5 points the x? and y? terms are 
dropped, and with 3 points only the linear terms are used. 
If a moving surface method is to be used for all interpolations, it must 
produce a surface that is continuous in areas where the terrain has this property. 
That can be achieved only by using the same formula for all interpolations. 
Further, to keep the computation time within reasonable bounds, only reference