-11 . 
boundary at these nodes. As a result, the two polynomials are identical and the 
total polynomial surface is continuous. 
Further, along a common boundary z-constant, ^, is in each local surface 
a third-degree polynomial in y. These two polynomials are similarly completely 
determined by the two values of A4 and the two values of their derivatives A4, at 
the nodes at the ends of the boundary. As a result, these two polynomials are 
identical. In other words, the two local surfaces have the same values of A, along 
their common boundary. The same reasoning applies to the values of hy along a 
boundary y=constant. Consequently, the total surface is also smooth. 
A bicubic polynomial is used also in the earlier described interpolation 
method of De Masson d'Autume [16]. In that method, the matrix A used in eq. (6) 
is the triple matrix product of eq. (5b). The matrix H in that product is con- 
structed from the heights at the nodes only. In one of the suggested simplifi- 
cations only the heights at the 4x4 nodes surrounding a grid element are used. 
With the stated arrangements of the heights in the matrix H, one has here 
A = DHD' (9a) 
If the origin of the local coordinate system is taken at the centre of the grid 
element and the sides have unit length, 
-9 69 69 
10 -150 150 (9b) 
36 -36 -36 36 
-40 120 -120 40 
It is, of course, not necessary to base this method upon the use of cubic 
splines through four points. If, instead, single cubic polynomials are used, 
eq. (9a) remains valid with only a different matrix D. A derivation similar to 
the one used for eqs. (7) and (8) shows that in this case 
3 27 27 - 
2 -54 54 -2 
12 -12 -12 12 
-8 24 -24 8 
Ds / 48 
In both cases, the total surface is continuous but it is not smooth at the 
boundaries. 
6.2 A l2-term bicubic polynomial 
The next simpler interpolation is by means of a 12-term incomplete 
bicubic polynomial obtained from eq. (6) by setting 
azz = azz = Q32 ? a33 * O (10) 
This polynomial may also be regarded as a quartic polynomial with three terms 
omitted. It is used by Jancaitis and Junkins [38] for local interpolation after 
the heights and slopes at the grid corners have been computed by the moving surface 
method. The four heights and the eight slopes at the corners of a grid element 
are just sufficient to compute the parameters of the local polynomial and to make 
the total surface continuous. However, the total surface is smooth only at the 
nodes, because only there the slopes are enforced in the adjoining local surfaces. 
If at the grid points only the heights are known, the slopes can be 
computed from the heights at the surrounding nodes. Leber] [8] does this in two