24 
u = C x + su + cv - bw + e (+u 2 -v 2 -w 2 ) + f.2uv 
+ p.2wu + 
cu + sv + aw + f (-u 2 +v 2 -w 2 ) + e.2uv + g.2vw 
+ 
W = c z + bu 
av + sw + g (-u 2 -v 2 +w 2 ) 
+ f.2vw + e.2wu + 
4.41 
Equation 4.41 represents a three-dimensional transformation from the 
(u, v, w) system to the (U, V, W) system in which the two-dimensional 
projections on each of the planes, u-v, v-w, and w-u, are transformed 
conformally. This can easily be verified by considering, for instance, the 
transformation between (u, v) and (U, V) planes. Substituting for w 
by zero in the first two equations of 4.41 and rearranging, one gets: 
U = c x + su + cv + e (u 2 -v 2 ) + f.2uv + ... 
4.42 
V = Cy = cu + sv - f (u 2 -v 2 ) + e.2uv + ... 
which is the more familiar two-dimensional conformal transformation. 
Equation 4.41 is, in addition, a direct extension of equation 
4.36 to include second-degree terms. The first four terms to the right of 
the equal sign of 4.41 are nothing but those of equations 4.36. This 
fact offers an advantage when the equations are programmed for the computer. 
One can formulate the least squares solution in such a way that the linear 
portion of the equations is applied first. A check is then made on the 
magnitude of the deformations exhibited to determine whether the second and 
higher-degree terms are required. Should the deformations be large enough 
to require higher-degree terms, the computer program automatically borders 
the matrix evaluated for the linear terms by the additional terms.