distance of a straight line. It remains to determine what constitutes a "negligible 

 distance" of a data point from a straight line, and to exploit methods for dividing 

 a profile data set into subsets each of which is appropriately identified with a 

 straight line. 



PROBLEM SOLUTION 



A. Graphical Solution 



The problem of BT profile reduction, stated in the last section, may be 

 regarded as a graphical problem of fitting lines to points in a plane. The metric 

 of the plane is defined by the dimensions of the standard BT slide scale. It is 

 assumed that a stylus trace on this plane will be of constant thickness independent 

 of its rotation in the plane. And it is assumed appropriate to fit a straight line 

 to points that align within the thickness of a stylus trace. 



It may be seen that the depth/temperature grid of the standard BT is 

 not linear (Figure 2). A "solution plane" may be defined in which temperature 

 and depth are related by a rectilinear grid in proportions that approximate the 

 BT grid. The grid of the solution plane is adequate for representing stylus trace 

 thickness as a constant independent of rotation. 



B. Tolerance a Function of Trace Slope 



Accordingly, a zonal section of the trace will vary in magnitude with 

 the slope of the trace, and points falling within a trace thickness on the plane 

 may be separated by a distance exceeding trace thickness on a zonal section 

 (Figure 5(a)). If a line is computed for least squares fit with points on the 

 trace, then the extreme zonal departure of a point from the straight line may 

 be permitted to vary with the slope of the line. Referring to Figure 5(a), the 

 permitted range of departure is ie, where 



e = d/(2sine). (1) 



This is translated to units of depth and temperature as follows. Suppose that 

 one inch on the plane of the BT slide equals kl intervals on the depth scale 

 and k2 intervals on the temperature scale. Then, referring to Figure 5(b), 



-e = '/yi^(i-¥^ 



(2) 



If the permitted range of departure on a vertical trace is expressed in units of 

 temperature, gT, so that e = ST/(2 sin 0), then the equation (1) may be 

 expressed 



