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ISPRS Commission III, Vol.34, Part 3A ,,Photogrammetric Computer Vision“, Graz, 2002 
  
THE IMPACT OF CONFORMAL MAP PROJECTIONS 
ON DIRECT GEOREFERENCING 
C. Ressl 
Institute of Photogrammetry and Remote Sensing, TU Vienna 
car@ipf.tuwien.ac.at 
KEY WORDS: Aerial Triangulation, Impact Analysis, Mathematics, Orientation, Calibration, Restitution, Algorithms 
ABSTRACT: 
If an aerial-triangulation (AT) is performed in the National Projection System (NPS), then the conformal map projection (which is 
the base for the NPS) effects the results of the AT by raising three problems: P1) the effect of the Earth curvature, P2) the different 
scales in plane and height, and P3) the variation of the scale in plane across large areas of interest. Problem P1 may be solved by the 
so-called Earth curvature correction. Whereas P2 and P3 have negligible effects in plane and height (at least for non-mountainous 
areas) when performing a conventional AT using control- and tie-points (so-called indirect georeferencing), their effect in height is 
not negligible (even for plane surfaces) when performing direct georeferencing (using GPS/INS). Simple solutions for solving these 
problems are presented — by considering the varying planar scale with the point heights or the principal distance. 
1. INTRODUCTION 
One of the main tasks in Photogrammetry is object 
reconstruction with a set of (aerial) photographs. The first and 
most important step for this task is image orientation; i.e. the 
determination of the images' exterior orientation (XOR). The 
interior orientation (IOR) is generally given by means of the 
protocol of a laboratory calibration. Afterwards in a second step 
(image restitution) object points may be determined by means 
of spatial intersections. Both steps (orientation and restitution) 
are often combined in the term georeferencing. 
The orientation part of georeferencing can be performed in an 
indirect way (aerial triangulation (AT) using ground control 
points (GCPs) and tie points (TPs); e.g. [Kraus 1996, 1997]) or 
in a more direct way (using satellite positioning systems (e.g. 
GPS) and Inertial Navigation Systems (INS); e.g. [Cramer 
2000], [Colomina 1999], [Skaloud 1999]. Since the 
photogrammetric relations are based on Cartesian coordinate 
systems, also the georeferencing should be performed in such a 
system, e.g. an appropriate chosen tangential system. However, 
in the end the reconstructed object points commonly need to be 
related to the National Projection System (NPS), which is 
commonly based on a conformal projection of the National 
Reference System (NRS). Therefore, the whole georeferencing 
process is often carried out already in the NPS. 
Since the NPS is based on a projection of the curved earth 
surface, during which distortions occur, the question arises, 
which errors are thereby introduced in the determined object 
points during the respective direct and indirect georeferencing. 
Concerning indirect georeferencing this problem was already 
discussed in the literature; [Rinner 1959], [Wang 1980]. The 
discussion for direct georeferencing will be presented in this 
article. Since only the point errors due to the NPS’ distortions 
are of interest, ellipsoidal heights will be used in the NPS.! 
  
! Actually in practise, instead of ellipsoidal heights orthometric 
heights are used in the NPS. Since in this case, the NPS’ height 
reference surface (i.e. geoid) differs from the NPS’ location 
Further all observations (image and GPS/INS) are considered 
free of errors. Especially, it is assumed that the principal 
distance during the image flight does not differ from the value 
determined during the laboratory calibration. 
2. THE CONFORMAL MAPPING OF THE NATIONAL 
SPATIAL REFERENCE SYSTEM 
As it was already mentioned, the object points determined 
during georeferencing need to be related to a given coordinate 
system, which in general is a well defined conformal projection 
of the National Reference System (NRS) - the National 
Projection System (NPS). One part of the NRS is an appropriate 
ellipsoid which is used as an approximation for the Earth’s 
shape (described by the geoid). By means of this reference 
ellipsoid 3D points can be described using their geodetic 
latitude (¢) and longitude (A) and their ellipsoidal height (H). 
The mapping of an ellipsoid can never be entirely true in length, 
only true in angles (conformal) or true in area. Due to the 
importance of (terrestrial) angular measurements most national 
reference systems are based on conformal mappings (e.g. 
Gauss-Krueger (or Transverse Mercator)). 
The Gauss-Krueger projection has the following properties: 
e The projection is conformal. Due to a break in the series 
development, however, this property does not hold 
exactly. 
e Only small stripes (1.5?) to the west and east of the central 
meridian (with %° overlap) are used for a single 
projection. 
e The central meridian is mapped in true length and serves 
as the ordinate axis of the strip system (Y wj). 
  
reference surface (i.e. ellipsoid) by the so-called geoid 
undulations, additional errors during georeferencing may be 
induced. These errors, however, are of physical nature and 
occur independently of the (geometric) errors induced by the 
map projection. Therefore their treatment lies beyond the scope 
of this work and only ellipsoidal heights are considered 
throughout this paper. 
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