as a regression function discontinuous in the first derivative at the points of 

 intersection between neighboring lines. Some of these points may subsequently 

 be deleted from the set of regression points finally selected to represent the 

 profile. 



E. Smoothing 



It happens that the tentative set of regression points may contain 

 virtually colinear points. That is, occasionally a point in the set may be a 

 negligible distance from a straight line connecting two points on alternate 

 sides of it. This point may be deleted and a single regression line may be 

 computed to fit the combined line sets on either side of it, provided the re- 

 sulting union of regression lines does not deviate by an intolerable distance 

 from any data point. It would therefore seem advisable to provide definitions 

 of a "negligible distance" between a regression point and a straight line, and 

 an "intolerable distance" between a data point and a regression line, as 

 criterion for smoothing a set of regression points. 



Accordingly, a "three level tolerance algorithm" (Figure 7) has been 

 devised for reducing a profile data set to a select set of regression points. The 

 first level of tolerance is used to define a "thin line", for assembling an initial 

 set of thin line sets to fit the profile data. The second level provides the 

 definition of a negligible distance between a regression point and a straight 

 line. The third level provides the definition of intolerable distance between a 

 data point and a regression line, A computer program (appendix) was developed 

 to analyze a profile data set and smooth the resulting set of regression points by 

 the trial and error methods outlined here, using the somewhat arbitrary ratios 

 1: VT: 2 to represent the three levels of tolerance. The union of regression 

 lines derived with this program does not deviate from the digital data profile 

 by more than the thickness of a "thin" stylus line (±2e). 



F, Width of the "Thin" Stylus Line 



Initial estimates provided by actual BT trace measurements for stylus 

 trace thickness were refined according to the following scheme. 



Several BT profiles were visually digitized to serve as data for the 

 development of the profile analysis algorithm and program. At a later time, 

 without reference to the initial data aggregates, the same profiles were 

 visually digitized a second time. Differences between temperatures at 

 corresponding depths in corresponding aggregates were calculated, individual 

 temperature differences were corrected for trace slope to provide an estimate 

 of the component of difference normal to the stylus trace in the solution plane. 



12 



