- 13 - 
from a cubic polynomial through the four points and from the mean of two quadratic 
polynomials, each using one outside point: 
h * (-hi*9h539h 4-h,)/16 (12a) 
The use of De Masson d'Autume's spline function would give: 
h = (-9h,+69h2+69h 3-9h, )/120 (12b) 
6.4 The 4-term bilinear polynomial 
Next in order of simplicity comes the popular interpolation by means of 
a bilinear polynomial: 
ago @o1||! 
h = [1 Jr I (13a) 
210 ail : 
This polynomial gives a fit at the four corner points of a grid element and it 
interpolates linearly along the boundaries and along all profiles parallel to the 
coordinate directions. These features can be used directly for the interpolation. 
Alternatively, selecting a local coordinate system in which the coordinates of the 
four corners are 0 and 1, and with an obvious numbering of the points, the 
interpolation can be written, see Schult [64]: 
h=nh + (h2-h1)æ + (ha-hi)y + (h1-h2-h3+hy )xy (13b) 
This interpolation is used in the British Integrated Program System for Highway 
Design, see Grist [6], by Gottschalk and Neubauer [10-2], and as an option in the 
IAGB model, see Benner and Schult [19]. It is discussed by Ehrlich [27], and it is 
one of the interpolation methods investigated by Leberl [8]. 
6.5 Double linear interpolation 
A computationally not obviously simpler interpolation in a rectangular 
grid is by double linear interpolation. This interpolation, also, has been 
investigated by Leber] [8]. It is obtained, although it is not usually formulated 
in this way by computing the height of the point at the centre of a grid element 
as the mean of the heights of the four corner points, followed by linear inter- 
polation in each of the four triangles formed by the sides and the two diagonals. 
The most primitive interpolation method, used as an alternative in [50], 
consists in linear interpolation in each of the two triangles into which a grid 
element can be divided by one of its diagonals. 
7. Interpolation in a net of triangles 
  
The fifth group of methods uses the reference points as the vertices of a 
net of triangles which cover the interpolation area without overlapping. If the 
reference points have been measured in an irregular pattern, the triangles will 
have irregular shapes. If they have been measured along parallel profiles in a 
. photogrammetric model and in adjoining profiles the points have been offset by half 
their distance in the profiles, a net of isosceles or even equilateral triangles 
will result. 
A continuous representation can here be obtained simply by linear inter- 
polation in each triangle. The interpolated height of a point in one of these 
triangles follows then from the condition that the volume of the tetrahedron which 
has the triangle as its base and the interpolated point as its apex must be zero: