Therefore the complete relations between the coordinates of the
fiducial marks in the frame and in the image are
X = X 0 +
J x cos<p + 3; sinw sin?> + h coso sin?> tan?>J
1 X sin?) + y Sino COS(p + h COS«) cos<p COSw)
3; coso — h sin«» )
c co Sor
+ tan<p, c sinor
F- F 0 +
\— X sin?> + 3; sin«) COS?> + h COS«) COS?)
[ x cos?> + y sin«) sin?> + h cos«) sin?) tan«)|
I X sin?) + y sin«) COS?) + h COS«) COS<p COS«)!
C Sinor
+
31 cos«) — h sin«)
: : : ; : -f tan«) | c cosor
X Sill?) + 3; Sin«) COS if + h COS«) COS?) J
Each of these expressions are expanded as a Taylor’s series with the
derivatives taken at zero for the rotations and at h for the principal
distance. As the deviations from these values are always small it is
only necessary to consider the first order terms to obtain the following
differential formulas:
X^
xy
dX = dX 0
+
dr ~
h
do) —
y da
+
xy
y~
0
II
+
~h d * -
h
de» +
X da
+
— X image % frame < d4 — ^ image y frame •
X
h
y_
h
dc
dc
As there are redundant observations the system of equations is solv
ed according to the method of least squares which, for the fiducial
marks shown in fig. 1, gives:
dX 2 + dX 4 dY + dY 3
dX0 = , dY 0 =
da = — 2 + b2) [a(dY! - dY 3 ) - b(dX 2 - dXJ]
h
dip = ( dx i — dX 2 + dX3 — dX 4 )
h
dc0 = — (dY t — dY 2 + dY 3 — dY 4 )
dc = 2 \J+ b2) [a(£/Zl ~ dXs) + b(dY * ~ dY ^ ]
471