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International Archives of the Photogrammetry, Remote Sensing and Spatial Information Sciences, Vol XXXV, Part B5. Istanbul 2004 
  
the local coordinate system with respect to the global coordinate 
system. 
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/ a 9 /V Lr X 
Z 
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Figure 7: Observed points in a plane 
The equation for the fictitious observation of the z-coordinates 
follows directly from (1) and is given below: 
z=0=z, +r (X-X)+rm,(Y=-Y) +r, (Z-2,) (6) 
The unknowns are: the object point coordinates X,Y,Z and the 
orientation of the local coordinate system: the latter comprises 2 
rotations (the rotation around the z-axis can be chosen 
arbitrarily). Further unknowns are: z, and the global coordinates 
of object points lying in the plane. 
Observations are: z-coordinates (70) of the points lying in the 
plane and the observations needed to determine these | 
unknowns. 
In case the fictitious observations are straight lines, the strategy 
would be to define two planes in that local coordinate system, 
which intersect in the desired straight line! 
The observed points lying on this line need not to be 
homologous. A thorough description is given in Kraus (1996) 
and will not be discussed here. 
3.1.3 Observed circles 
A circle in object space can either be described through an 
intersection of a sphere or a cylinder with a plane. This means 
that feature points have to lie both in the plane and on the 
surface of the sphere or cylinder. Here the advantage of using 
local and global coordinate systems for these observations 
becomes clear. The cylinder or sphere as well as the plane are 
analytically described in a common local coordinate system 
(Figure 8 and Figure 9). 
ZA 
  
  
e 
Figure 8: Fictitiously observed circle using a cylinder 
  
* Ya“ ^Y; 
  
X 
Figure 9: Fictitiously observed circle using a sphere 
Implicit function ofa sphere: $202 x? 4 y-v-z-p ON 
Implicit function ofa cylinder: C 20 2 x? 4 y-r (8) 
Explicit function ofa plane: z 20 (9) 
x,y and z are the coordinates of the adjusted point P after a 
spatial congruancy transformation from the global coordinate 
system into the local coordinate system: 
xeA Y (10) 
X, y and z in the observation equations (7)-(9) must be 
substituted by the quantities Y , Xo and R in relationship (10). 
The unknowns are the three translation parameters and the two 
rotation parameters of the local coordinate system, as well as 
the radius of the cylinder. The rotation around the z-axis can — 
and has to be chosen arbitrarily. 
Observations are the z-coordinates (z=0) of the points of the 
plane and the zero-distance of the points from the sphere or 
cylinder which is subject to adjustment as the algebraical or 
normalized residuals (Kager, 2000), as well as the image 
coordinates of the observed points. 
4. BLOCK SCALING 
As mentioned before, the only scaling information available 
were the brake disk diameters. In order to use this information, 
the brake disks had to be modelled (see chapter 3.1.3). Again, 
no homologous points could be found on the brake disk 
circumferences, hence fictitious observations had to be 
introduced. Two geometric features were described for each 
disk: a sphere and a plane (see Figure 9). 
The plane was defined through all the observed points on the 
brake disk. The sphere was defined through points lying on the 
brake disk circumferences. The intersection of these two 
features lead to the circles in space that corresponded to the 
outer brake disk brink. 
The radius r (Eq. 7) was not considered as unknown, but it was 
given a fixed value: the brake disk’s radius from the 
motorcycle’s specification sheet. It was important to set the 
previously defined range (chapter 2.2) between the two 
arbitrarily chosen points free; so there would be no scale-over- 
parameterisation of the block. After the final adjustment the 
block got its final, correct scale. 
5. FORK MODELLING 
Now, that the image block was properly oriented and scaled, the 
actual work for computing the angle of the break of the front 
fork could begin.