ISPRS Commission III, Vol.34, Part 3A „Photogrammetric Computer Vision‘, Graz, 2002 
Sketch (A) in Figure 1 shows the section through three 
projection centers (PRCs) of a strip (flown from West to East) — 
for simplicity reasons the section ellipse is drawn as a circle. 
The plane flies in constant ellipsoidal height Hy. The principal 
distance is c and the image format is s. If the Gauss-Krueger 
projection is applied to the ellipsoidal area covered by these 
images, then this area is unwrapped in a conformal way (so to 
speak). The length distortions introduced this way shall be 
neglected at first. This projection of the ellipsoidal surface 
delivers the planar coordinates for the NPS. The Zmap 
coordinate is made by the ellipsoidal heights of the surface 
points, which are related to the curved reference ellipsoid of the 
NRS. This ellipsoidal curvature prevents the direct usage of the 
Photogrammetric Cartesian relations in the NPS. 
In a first order approximation the curved surface of the ellipsoid 
can be replaced by a polyhedron of tangential planes, with each 
plane set up at the nadir point of the PRCs. Then the 
unwrapping of this polyhedron gives a first order approximation 
for the Gauss-Krueger projection. This together with the heights 
related to the respective tangential plane for each image create a 
(small) individual Cartesian system of coordinates. The 
correction needed for this tangential approximation therefore 
removes the effects of the ellipsoidal curvature. 
Sketch (B) in Figure 1 depicts this correction. There a 
meridional section of the area around the nadir point T; of an 
image i together with its tangential plane and the curved surface 
of the ellipsoid is shown. Further a point P; on the Earth surface 
is depicted, having the coordinates (Xr, Yr, Zt) with respect to 
the Cartesian tangential system. If the NPS coordinates of T; 
and of all points P; that are observed in image i, are 
(approximately) known, then these points can be transformed 
into this individual tangential system. In this system the 
Cartesian relations of the Photogrammetry do hold. 
This transformation is commonly termed as Earth curvature 
correction and it is a standard module in today’s AT packages. 
This correction can also be performed in the way, that the object 
coordinates are not altered but the image coordinates. For the 
Earth curvature correction the reader is directed to e.g. [Wang 
1980], [Kraus 1996, 1997]. 
By means of this correction the flight in constant ellipsoidal 
height Hp is flattened; i.e. now the plane flies horizontally in 
constant height Hg above the reference plane used for the un- 
wrapping of the tangential polyhedron; Sketch (C) in Figure 1. 
Now the previously neglected length distortion t of the Gauss- 
Krueger projection together with its increase with the distance 
from the central meridian and the unchanged usage of the 
ellipsoidal heights in the NPS are taken into account. These 
three circumstances introduce a contradiction, which is depicted 
in the Sketches (D1) and (D2) in Figure 1: On the one hand, 
since the plane flies in constant flight height, also the PRCs in 
the NPS should have the same Zy, coordinate. This, however, 
induces an increase in the view angle because of T and its 
increase to the East; i.e. the ratio between principal distance c 
and image format s must change continuously (Sketch (D1)). 
On the other hand, if this ratio is kept constant (since all images 
are taken by the same camera), then the Zmap coordinate of the 
PRCs must increase to the East (Sketch (D2)). 
To sum it up, three problems occur if one wants to perform an 
AT in the NPS: 
P1) the effect of the curvature of the Earth 
P2) the difference between planar and height scale 
P3) the continuous change of the planar scale throughout 
the considered area of interest in lateral direction 
Whilst problem P1 can be solved using the afore mentioned 
correction of the Earth curvature, the other two problems have 
been neglected so far — at least to the knowledge of the author. 
The question now arises, which errors are induced in the 
determined object points during direct and indirect 
georeferencing, if the problems P2 and P3 are neglected. 
Note: For terrestrial geodetic (polar) networks computed in the 
NPS sometimes the so-called arc-to-chord correction needs to 
be applied. This correction compensates for the angular 
deviation between the straight line connecting two points in the 
map of the NPS and the (curved) map of the straight line 
connecting the respective points on the earth surface. For 
photogrammetric networks, however, this reduction can be 
ignored, since the effect of this reduction referred to the image 
is less than 1 um: For the Gauss-Krueger projection (A — 1.5? , 
© = 48°  Xys, ~ 112 km) if the image scale is larger than 
1:42.000 and for the UTM projection (4 —3.0?, 9-48? > 
Xwmap — 220 km) if the scale is larger than 1:21 .000. 
3. INDIRECT GEOREFERENCING IN CONFORMAL 
MAP PROJECTIONS 
In this case the following quantities are given: 3D GCPs (in the 
NPS), the coordinates of their mappings in the aerial images, 
the image coordinates of TPs and the IOR according to a valid 
calibration protocol. The parameters to be determined are: the 
images’ XOR and the 3D coordinates of the TPs (and of other 
spatial objects in the subsequent stereo restitution). Problem P1 
is solved using the Earth curvature correction. What about P2 
and P3? 
During the orientation step the XOR of the images is indirectly 
determined using the GCPs and TPs. The free flight height 
adjusts to the GCP and TP situation on the ground; i.e. the local 
planar scale — caused by t and realized in the GCPs — is 
transferred into the flight height. This situation is depicted (two 
images, one GCP and one unknown point N, which shall be 
determined during image restitution) in Figure 2. 
Cartesian: NPS: 
AT A 
He X A DON 
T N (GCP ci | N le: 
{Hy | HN fes 
m ETE 
Figure 2: Indirect georeferencing 
The pencil of projection rays is congruent for both systems (the 
Cartesian one and the distorted one, i.e. the NPS), since in each 
system the same value for the principal distance c is used. 
Therefore it holds: 
AH p » (r-1) (Hp - Hacp) (2) 
A - 285