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First of all, the relation between dz, and the distortion in the projection plane is:
R
dR = — È dz,
where dR is the distortion at a distance E from the point 33, h is the projection distance
and dz, the determined quantity (27) or the corresponding quantity from other sets of
points. The distortion dR is assumed positive if projected points are displaced in the
direction from point 33. See Fig. 1 and 2.
At the projection distance f we have the distortion as
(38)
Rf
dr = — he dz, (39)
r
or dr = — X dz, (40)
where r is the radial distance in the grid.
Inserting the expression for dz, from (27) into (40) we obtain
rf
dr = Sab Ng, (41)
In the example above we have » =a V2 and the expression Ny, in (21).
Hence we find in this case:
f v2
dr = u^ —g Cdi t dy, +dyss— dæ,, +dæ,s —dx,, +dxss) (42)
If photography is assumed, the distortion has reversed sign.
The standard error of dr is directly found as:
f
m, = 2h HU (43)
where u is the standard error of the measurements, determined from (37).
The same procedure can be applied to all other combinations of points, along circles
around 33.
Below we assume the five possible combination of points according to Fig. 2. The
radial distances from point 33 are:
a a a V5 i.
q' yg 9 V 2. We denote the radii r, — 7.
From the solution of the normal equations we obtain the following expressions for the
distortion:
f
f V2
dr, = TEL. (— dias — dy3, + dY49 * dy,, — di, -- dz, — dX42 + days) (45)
f
f V5
dr, = wot 2dy 1p — 2dy,, — dy, — dyos + dy4, + dyas - 2dyg, - 2dyg, —
— dæ,9 + dæ,4 — 2dx9, + 2dx,5 — 2dx,, + 2dæ,5 — dæ59 + dx54) (47)
f V2
dry ——gh (— dyır — dYı5s + ds, + duss — dæ,, + des — dag, + dogg) (48)
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