4 
I 
14 
where 
(X) — (X—Xq) cos op cos * + (Y—Y 0 ) (sinco sinç? cos* + cosoj sin*) — 
— (Z—Z 0 ) (sinco sin* — cosoj sin^? cos*) 5.1.3 
(Y) = — (X—X 0 ) COSCp sin* + (Y—Yo) (cosco COS* sinoj sin<£> sin*) — 
— (Z—Zq) (cosoj sinç? sin* + sinoj cos*) 5.1.4 
(Z) = (X Xo) sinç? + (Y—Yq) sinoj coscp + (Z—Zq) COSOJ cosy> 
5.1.5 
X Y Z: co-ordinates of points in the object 
x y : co-ordinates of points in the image 
XqY 0 Zq: co-ordinates of exterior center of projection 
: co-ordinates of principal point 
c : principal distance (camera constant) 
o) çp * : rotation of the picture 
5.2. RADIAL DISTORTION 
The aberrations of a lens composed of spherical surfaces can be computed 
from Seidel sums. There are five and the fifth sum gives the radial distortion. 
This is the aberration which is most important for photogrammetry. From these 
aberration calculations the radial distortion can be shown to be an odd powe 
red polynomial with tan Q as the independent variable. Here 0 is the angle 
from the optical axis to the ray and if the image plane is perpendicular to this 
axis we can also write 
dr = ogr 3 + fl-jr 5 + a-j 1 + « 9 r 9 + . . . . 5.2.1 
where 
dr : radial distortion 
r = ] (x—x 0 ) 2 + (y—y 0 ) 2 
at : constants. 
If the constants m are calculated from the Seidel sums, the radial distortion 
refers to the Gaussian focal length. From similar triangles in Fig. 4 we obtain 
5.2.2 
r c 
Expression 5.2.1 can be converted to 
dc f = c a^r 2 + c a 5 r 4 + 5.2.3 
which expresses the radial distortion as a varying camera constant. This is a 
correction to the camera constant, c, in the basic formulas 5.1.1. and 5.1.2. Com 
bining formulas 5.1.1, 5.1.2 and 5.2.3 we obtain the change in the image co 
ordinates as