17 
If formula 5.4.4 is combined with 5.4.5 and 5.4.6 we obtain 
5.4.7 
5.4.8 
Ax — — d sin a (—(x—x 0 j sina + (y—y 0 ) cos a) 
Ay = S cos a (—(x—x 0 ) sina + (j—Jo) cosa) 
If we want to have unit scale in ^-direction, i.e. Ax in formula 5.4.7 to be zero, 
we have to change the scale uniformly by an amount — d sin 2 « and rotate 
the system by an amount — d sin a cos a. After these corrections of formulas 
5.4.7 and 5.4.8 we obtain 
dx = 
Ax — ò (x 
—Xq) sin 2 a + d 
(j- 
-yo) 
sina 
cosa - 0 
5.4.9 
dy = 
Ay — d (y 
—y 0 ) sin 2 a — d 
(x- 
-*o) 
sina 
cosa = 
— 
ò (cos 2 a — 
sin 2 «) (j—Jo) - 
- 2 < 
3 (x- 
-Xo) 
sina cosa 
= 
(y—yo) dm + (x—Xq) dp 
5.4.10 
where 
din = 
ò (cos 2 «—; 
• o \ 
sinraj 
5.4.11 
d^ = 
— 2d cosa sina 
5.4.12 
>x 
Fig. 5. 
Affine shrinkage. Image co-ordinates are measured in the x-y-system. There are two scales 
of the image co-ordinates caused by a shrinkage of the picture in the direction of the t-axis. 
Putting the scale along the j-axis equal 1, we obtain a scale of 1 + <5 along the ¿-axis. This 
causes generally a scale difference and a lack of perpendicularity in the image co-ordinates 
measured by the x-y-system.