16 
R+D*É*Â*b=r (5) 
All bold letters denote a vector, the roof denotes a matrix. D is the scale factor, D*b is the distance from the centre of 
projection to the projected point in the projection plane. The scale factor D and the horizontal components of r,r, and r,, 
can be calculated if  is known. À depends on three, in aviation well defined Euler angles (Schulz/Ludwig, 1954). The 
diagonal matrix É turns the camera fixed co-ordinate system into the Earth fixed co-ordinate system. The product E*A 
changes only the sign of the lower two rows of A. The z component of R is the altitude H. The attitude angles make the 
scale factor D pixel depending: 
D=D= (H-z,,) /( d*À, + i*dp *A + * A.) (6) 
The position of imaged points in the projection plane are: 
r, =R, + D,*[d*A,, +i*dp *A + f* A ] 
^ 
r, =R,+ D '[-d*À, - i'dp *À,- f* ÀJ (7) 
These equations are called the collinearity equations. But they are not sufficient for on- line photogrammetry. It is 
necessary to know the velocities of the imaged points in the projection plane. The equation (5) is differentiated by time: 
d/dt r = d/dt (R + D*E*A*b) (8) 
On the right side R depends on the time because of the linear velocity of the airplane. The scale factor D and the 
matrix À depend on the Euler angles, which are changing because of the angular velocity. 
d/dt r = d/dtR + (d/dt D)*E*A*b + D*E*(d/dtA)*b (9) 
d/dtR=v, 
The last term (9) can be written using the measured angular velocity vector: 
v, = © x D*b = D*(@ x b) = D*0*b (10) 
- 0 -Q, Q, 
Uz Q, 0 -, 
-® , 0 
So this linear velocity component transformed to the Earth fixed co-ordinate system is : 
v,=D*E*A*0D (11) 
The second term (9) can be calculated using the z-component because this component has to be zero: 
ddr - d/dt R, -d/dt D *(d*A,, + i*dp*A,+f*A,.) 
-D* [d*(0,‘A,, - ©, *A,,) + l*dp*(0,*A,, - @,A,,) +f*(0,*A, - OA) = 0 ! 
d/dtD = d/dtR -D* [d‘(o'A, - © A,) + i*dp (0 A, - 0 *A,,) +f*(0 A, - 0A) = V, 
(d*A,, + i*dp*A,,+*A,,) (12) 
The two horizontal components of the linear velocity in the projection plane are: 
Var = Vat Voy T Vay (13) 
The last two terms depend on the pixel position i*dp along the sensor line and so the distances between different 
projected pixels are different. On the other hand the projection of a straight line is again a straight line. These pixel 
depending velocities produce the dilatation. Different projected lines loose their parallelism. The Figure 2 shows the 
traces of some ground pixels in the projection plane. A curved line from bottom to top is the trace of one pixel (centre). 
The straight lines are the projected sensor lines (centre). The time resolution is 1 ms and we have calculated the 
image for every 100th pixel and have plotted the projected line every 100 ms using real attitude data and the WAOSS 
parameter. So, the mesh size is given by v*100ms in flight direction and by the varying distances of 100 projected pixels 
in line direction. It was calculated using the time depending collinearity equations. The time depending calculation of A 
is possible, if the relation between the measured vector o and the Euler angle velocities is known. Bold printed Euler 
IAPRS, Vol.30, Part 5W1, ISPRS Intercommission Workshop "From Pixels to Sequences", Zurich, March 22-24 1995 
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