D. Intersection 



Neighboring regression lines may or may not intersect in the range of 

 overlap. If they intersect, then the depth of intersection may be used to 

 represent the limit of regression lines on neighboring line sets. The two lines 

 will form a path, or "union", that will be within half a stylus trace thickness 

 of data points contained in the two line sets. If they do not intersect in the 

 range of overlap, alternative methods must be used to represent the union of 

 neighboring line sets. The path derived by extending the regression lines to 

 the point of intersection outside the range of overlap will fail to represent all 

 data points in the union of neighboring sets with predictable precision (Figure 6). 



Neighboring regression lines frequently do not intersect in the range 

 of overlap when their line sets do not overlap. Hence it is necessary to examine 

 alternative combinations of line sets that overlap when searching for a union of 

 regression lines to represent a profile. Thus, for example, when the regression 

 lines on the line sets (A, B, C) and (D, E, F) do not intersect between points 

 C and D, then an intermediate line set, defined for point C, must be considered. 

 Occasionally, when they are nearly parallel, the regression lines for overlapping 

 line sets will not intereect in the range of overlap. In this case the regression 

 lines must be close together in the range of overlap, since both lines must fall 

 within half a stylus line thickness of data points contained in the range. 



The following logic therefore seems to be suggested as a means for 

 selecting a union of lines to fit a digital profile: (1) Starting from the surface, 

 identify a "base" regression line, and select a neighboring regression line to 

 represent a union with the base line; (2) When a neighbor is selected, identify 

 the neighbor as a base line, and repeat the selection process downward, until 

 the set of regression lines is exhausted. Neighboring regression lines may be 

 examined and selected for the behavior of intersection with the base line 

 according to the following scheme: (1) If the point of intersection falls in 

 the range of overlap, tentatively identify it as a "regression point"; (2) If 

 the point of intersection does not fall in the range of overlap, determine whether 

 their line sets overlap. If they do not overlap, then the union of the base 

 regression line with its overlapping neighbor must be examined, repeating the 

 procedure above, if they do overlap then the median depth in the range of 

 overlap and the median temperature between the two regression lines at that 

 depth may tentatively be selected as a substitute regression point between the 

 two line sets. This point is within half a stylus trace thickness of the regression 

 lines on the two line sets, and constitutes a tolerable compromise. 



A set of regression points may thus be tentatively defined to represent a 

 union of straight lines fitted to a profile data set. The union may be regarded 



10 



