Discussion
The directions in the preceding paragraphs can be turned into a computer
program. However, a difficulty may arise when the program is asked to evaluate
the limit of the substituted expression as t→0. In that case, apply the following
artifice. Reset the precision of the calculations to 25 digits. Assign (t) as a small
number such as 10–18. Then set z = –1 to focus on figure ABDC. Evaluate the
substituted equation with a lower precision such as 10 digits. This artifice or a
similar one applies in many cases. The proposed assignments are suggestions.
The first column in Table 1 lists functions that generate the reference surfaces.
P is taken as (5+x+3y) for each function listed in the first column of Table 1. If
the function is 100/P, then the reference surface is 100/(5+x+3y). The coordinate
system is –1 .. 1 in both the x- and y-directions. The second in column lists sums
of squares of deviations of the bilinear equation from the reference surfaces. The
third and fourth columns list the analogous sums as obtained by the four-point
quadratic equations (A) and (B). See section 2. The fifth column lists sums of
squares of deviations based on the cubic equation model for the four-point
rectangle described in section 3. Table 1 compares four models for ABDC