-10 = 
to a corner or to the centre of each grid element, these linear functions have the 
same coefficients in all grid elements. The coefficients are, therefore, computed 
once for all in advance of the interpolation. In practice, that can be done very 
conveniently by writing the matrix A as a function of a matrix H which contains 
the heights and other values, if any, as its elements. 
6.1 The_16-term bicubic polynomial 
The most sophisticated method of the present group makes use of the full 
16-term bicubic polynomial given by eq. (6). As described earlier, Bosman et al 
[50], [22] make use of this polynomial. They derive its parameters from the values 
of h and of three derivatives of A^ at the four nodes of a grid element. The first 
derivatives hz and ^, express the slope of the surface in the x- and y-direction, 
respectively. These are obtained from eq. (6) by differentiating the vectors x 
and y with respect to the coordinates x and y, respectively. The mixed second 
derivative hy, is obtained by differentiating both vectors. From the values of A 
and its derivatives at the corners of a grid element, the matrix A can be computed 
as follows. The components of the vector x and of its first derivative are 
computed for the two values of x at the boundaries of the grid element and the 
four resulting column vectors are transposed and placed below each other to form 
a 4x4 matrix X. A matrix Y is computed similarly from the vector y. Arranging 
the four heights and their derivatives suitably as the elements of a 4x4 matrix H, 
the equations (6) for the four heights and the equations for their derivatives can 
be assembled in one matrix equation: 
H = XAYT (7) 
Choosing a local coordinate system in which the four corners of the grid 
element have coordinates 0 and 1, 
Here, Hy; 
X 
for the grid point with coordinates Z and j. The solution of eq. (7) is 
A= XT 
in which 
-3 
This is the matrix dng of Bosman et al if their grid sides have unit length. 
For each two adjoining surfaces, the profile at their common boundary is 
a third-degree polynomial with respect to either x or y, as the case may be. Each 
polynomial is completely determined by the two grid nodes at the ends of the 
boundary and the two first order derivatives or slopes in the direction of the