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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