ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision“, Graz, 2002 
  
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Figure 1: A flight strip in the NPS 
Only the nadir point Py of a given 3D point P with ellipsoidal 
height H is used during the projection of the point P. This 
results in the planar coordinates (XMap: YMap) Which are affected 
by the length distortion T of the map projection. T (and hence 
the planar scale) increase quadratically with the distance from 
the central meridian. To complete the 3D coordinates in the 
NPS the ellipsoidal height H is used as the height coordinate of 
the mapped point (Zy,, = H) — as a consequence, the skew 
normals of the ellipsoid will be mapped to parallel lines. Thus 
planar scale and height scale are equal only along the central 
meridian — whereas with increasing distance from the central 
meridian the difference between these two scales gets larger. 
Due to these reasons the NPS does not represent a Cartesian 
coordinate system. 
For the Gauss-Krueger projection the length distortion T in the 
lateral distance Xmap from the central meridian is computed in 
the following way — with R being the mean radius of curvature, 
depending on the reference ellipsoid (a, b) and the geodetic 
latitude  [Bretterbauer 1991]: 
  
  
2 4 
X X 
r-l1e— p. Mp (1) 
2R 24R 
2 uni 
1+e""-cos* @ b b 
The effect of t on 1000 m at one strip’s border is (A = 1.5°, Q- 
48° — XmMap - 112 km) ~ +15 cm and at the end of the overlap 
(A =2.0°% © = 48? 2 Xy, — 150 km) ~ +28 cm. 
Another widely used projection is the Universal Transverse 
Mercator (UTM) which is based on Gauss-Krueger but uses a 
strip width of 3°. To reduce the effect of the length distortion, 
the planar coordinates for this map are altered by the factor 
0.9996. In doing this the true length in the central meridian is 
lost, but is achieved in parallels to the Ywmap axis in ~ 180 km 
distance to the west and east of the central meridian. 
The effect of T on 1000 m is in the central meridian (À = 0.0°, Q 
= 48° — Xy — 0 km) — —40 cm, at one strip's border (A — 3.09, 
Q — 48? Xy, — 220 km) — +20 em and at the end of the 
overlap (A = 3.5°, 0 = 48° —> XMap - 290 km) ~ +65 cm. 
In this distorted system of the NPS the coordinates of the object 
points are to be determined given the aerial images. However, 
the equations used in Photogrammetry (e.g. for the central 
projection) and the points determined with them refer to a 
Cartesian coordinate-system. How can this problem be solved? 
The first (and theoretically best) method is to perform the AT in 
an Cartesian auxiliary system (e.g. a tangential system set up in 
the center of a given area of interest) and to transform the 
results to the NPS afterwards. This method, however, also has 
some (practical) drawbacks: 
a) Problems during the so-called refraction correction may 
arise (esp. for large areas of interest), since this correction 
usually assumes the plumb line direction to coincide with 
the computing systems' Z-axis, which is not rigorously 
valid in the tangential system. 
b) Due to the same reason certain leveling constraints (e.g. 
points along the shoreline of a lake, or the leveling of a 
theodolite if polar measurements are introduced into the 
adjustment) need some sophisticated realization in a 
tangential system. 
C) The most severe drawback, perhaps, is that the results of 
the stereo restitution (roofs, streets, natural boundaries, 
contour lines, etc.) are finally required in the NPS. But 
today's analytical and digital plotters (esp. the CAD 
module) do not fully support (at least to the knowledge of 
the author) the digitisation in the tangential system and 
the simultaneous storage of the results in the NPS. 
Therefore, in the second method, the AT is already performed in 
the NPS and the discrepancies between the distorted NPS and 
the Cartesian ‘nature’ of the Photogrammetric equations are 
minimized using suitable corrections. In the following the 
problems that arise with this second method are discussed in 
detail. Figure 1 depicts the situation schematically. 
  
  
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