One important feature concerning connections betweens the points in the two planes (, ) and (, ) is the fact that (with only a few exceptions that will be noted) straight lines in one plane transform into straight lines in the other. For example, points satisfying

= A + B
in the (, )-plane (where *A* and *B* are constant
intercept and slope, respectively), then satisfy

= A^-1 - A^-1B in the (, )-plane. So long as in (AB1), the straight line in (AB1) transforms into the straight line in (AB2). This observation is very important because the straight line in (patchy) corresponds to a straight line in the saturation-proxy plot in the (, )-plane. But this line transforms into a straight line in data-sorting plot in the (, )-plane. In fact the apparent straight line along which the data align themselves in these plots is just this transformed patchy saturation line.

When *A* = 0 in (AB1) [which seems to happen rarely if ever in the real
data examples, but needs to be considered in general], the resulting
transformed line will just be one of constant ,which is a vertical line on the (, )-plane.
The more interesting special case is when *B*=0, in which situation
or . But this case includes
that of Gassmann-Domenico for homogeneous mixing of the fluids at
low to moderate saturation values. For *B* = 0, on both planes we
have horizontal straight lines, but their lengths can differ
significantly on the two displays.

4/28/2000