Tsig 



From figure 4, the planar ridge truncation ranges from t = at S = 840, to t = 10 

 at S = 1240. This plane, scaled to 1.0, is matched and smoothed by Tsig, as shown in 

 figure 12. Note from the Tsig formula specified (p. 13) that the inverted sigmoid to 

 the left of the truncation plane (with base = 1.0 and peaking at 0, 0) was described 

 on the reversed S-axis (6000-S) and subtracted from 1 . to arrive at Tsig. This 

 provided a more accurate duplication of the truncator plane than did other sigmoid 

 alternatives, in this case. 



Although a maximum of S = 6000 appears in Tsig, the applicable range is still 

 limited to S = 5000 based on the original figure (fig. 4). 



"co 



PLANAR RIDGE 

 TRUNCATION SCALED TO 1.0 



1.0 

 .8 



.6 

 .4 

 •2L 



.0 



Tsig 



1 1 — W — 1 



840 1240 2000 3000 6000 (S) 



6000 5160 4760 4000 3000 (6000 - S) 



Figure 12. — Ridge truncation sigmoid, Tsig. 



Rsig 



The right segment of figure 4 has a constant cross-section over F for 1240 f_ S <_ 

 5000, approximated in the descriptor by a sigmoid oriented at RO and extending to the 

 right for a distance of 11 units in F. Eleven is about the minimum operational span 

 for the right-segment sigmoids. RO ranges from 8.404 at S = to 10.808 at S = 5000. 

 A range of 11 includes F = zero at both extremes, and so was adopted. The Rsigs were 

 described as a function of the F-transform | RO-1 1 | +F . . . as shown in figure 11. Thus, 

 the maximum value of 11 always occurred at the sigmoidal peak, F = RO, and Rsig 

 functioned over the range zero to 11 of the F-transform. 



The final surface, then, is simply the independent sum of the three contiguous 

 ridge segments ... al 1 truncated at appropriate points in S through multiplication 

 by the proportional values of Tsig. 



15 



