distance respectively. The relation between the grid co-ordinates and the 
corresponding co-ordinates in the projection plane is approximately: 
x'h 
yh (1) 
This relation would be strictly valid if no errors of the inner or outer orientation 
of the projector and of the measurements were present. 
But since there are small errors in the orientation elements and in the 
measurements we will find discrepancies in the measured co-ordinates of the 
projection plane with respect to the relation (1). 
If we denote the measured co-ordinates by (x) (y) and the ideal co-ordinates 
according to (1) by x, y, we define the co-ordinate errors by 
dx = (x) — x 
2 
dy = (yy e 
Since we have assumed small errors in the elements of orientation, dx and dy 
can be expressed by the well-known differential formulae: 
X xa Xy 
dx = dxo -t j^ —ydr E (1 = hr hdé «ir j do (3) 
y Xy y? 
dv — dva +— dzo + Xd + ~~ dé + ( + T) hdo 4 
) Yo h 0. A ji C (4) 
In these formulae the differentials are as follows: 
dx, = a translation of the projector in the positive x- direction; 
dy, = a translation of the projector in the positive y- direction; 
dz, = a translation of the projector upwards; 
d«  — a rotation of the projector around the projector axis, positive 
counter-clockwise as seen from above. 
Cà = a rotation of the projector around an axis through the projection 
centre parallel to the y-axis and positive counter-clockwise seen 
in the positive y- direction; 
de — a rotation of the projector around an axis through the projection 
centre parallel to the x-axis and positive clockwise seen in the 
positive x- direction. 
The errors of the inner orientation of the projector can completely be 
compensated by the elements of the outer orientation according to formulae 
(3) and (4) except for the radial distortion. 
For the determination of the six differentials of formulae (3) and (4) 
we need at least six observations (x) and (y) which correspond to three points. 
If more observations are available we write equations (3) and (4) as observation 
equations. If we furthermore want to determine the corrections dx, dyo, dz 
etc. we reverse the signs of all differentials and the observation equations for 
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