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THEORY OF OPERATION 
The theoretical basis for the comparator is best illustrated with the aid of 
Fig. 5 which shows the four measurements r, j1 Ia 47 Ta 47 T4 of apoint x, y,. 
It will be recalled that in the actual measuring process, the pivot of the measuring 
arm remains stationary while the plate is measured in four different positions. 
Upon reflection, one can appreciate that this is precisely geometrically equivalent 
to a process in which the plate itself remains stationary while the pivot assumes 
four different positions. The actual measurements r,, are from the zero mark of 
the scale to the point rather than from the pivot to the point. To convert the 
measurements to radial distances from pivots one must specify the radial and 
tangential offsets (y, g) of the pivot relative to the zero mark. If x^, y& denote 
the coordinates of the pivot corresponding to the i th position of the plate 
(i7 1,2,3,4), one can write the following four observational equations relating 
the measured values r,, and the desired coordinates x,, y ,: 
(yy +a)? + 7 = be, = * (y, X 
(2110 * 89* 7, X ry XS? 
(rs 3 +a)? + B2 = (x, 5G) + ly, -y5)? 
(ry +a)? + B2 = (x, =x§)2 + (y, - 5 
These equations recognize that since there is in reality only one pivot and 
measuring arm, a common @ and B apply to all four positions of the plate. If the 
ten parameters of the comparator (i.e., «and 8 plus four sets of x5, yz) 
were exactly known we could regard the above system as involving four equations 
in the two unknowns x,, y,. Accordingly, the process of coordinate determination 
in this case would reduce to a straightforward, four station, two-dimensional least 
squares trilateration. 
In practice, the parameters of the comparator are not known to sufficient 
accuracy to warrant their enforcement. lt follows that they must be determined 
as part of the overall reduction. This becomes possible if one resorts to a solution 
that recovers the parameters of the comparator while simultaneously executing 
the trilateration of all measured points. Inasmuch as one is free to enforce any 
set of parameters that is sufficient to define uniquely the coordinate system being 
employed, three of the eight coordinates of the pivots can be eliminated through 
the exercise of this prerogative. In Figure 5 we have elected to define the y 
axis as the line passing through pivots 1 and 3, thereby making x£ = x = 0. 
Similarly, the x axis (and hence also the origin) is established by the particular 
line perpendicular to the y axis that renders the y coordinates of pivots 2 and 4 
of equal magnitude but opposite sign (thus y£ ^ -y$). This choice of coordinate 
system has the merit of placing the origin near the center of the plate. 
By virtue of the definition of the coordinate system, only seven 
independent parameters of the comparator need be recovered. If n distinct points 
are measured on the plate, equations (1) may be considered to constitute a 
system of 4n equations involving as unknowns the seven parameters of the 
comparator plus the 2n coordinates of the measured points. When nz 4, there will