TWO-DIMENSIONAL GEODETIC MODELS - GPS Satellite Surveying - page 331

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positions can subsequently be mapped to the conformal plane. A user might not even
be aware that geodetic coordinates had been used in the adjustment.

9.2.1 Reduction of Observations

Fi gure 9.7 shows the mapping elements ( γ , ∆ t, ∆ s) that link the geodesic observa-

) to the corresponding observables on the mapping plane ( d,

tio ns (s,

¯

α t) . Note that

s is the length of the geodesic on the ellipsoid and not the length of the mapped
ge odesic that does not enter any of the equations below. The mapping plane must not
be confused with the local astronomic or geodetic horizon. It is simply the outcome
of mapping the ellipsoidal surface conformally into a plane. One can generate many
su ch mapping planes for the same ellipsoidal surface area.
In Figure 9.7, the Cartesian coordinate system in the mapping plane is denoted by
(x , y) . The points P 1 (x 1 ,y 1 ) and P 2 (x 2 ,y 2 ) are the images of corresponding points
on the ellipsoid. Consider for a moment the geodesic that connects the points P 1 and
P 2 on the ellipsoid. This geodesic can be mapped point by point; the result is the
m apped geodesic as shown in the figure. This image is a smooth but mathematically
co mplicated curve. The straight line between the images P 1 and P 2 is the rectilinear
ch ord. The image of ellipsoidal meridian may or may not be a straight line on the
m ap. In order to be general, Figure 9.7 shows the tangent on the mapped meridian.
Th e angle between the y axis and the mapped meridian is the meridian convergence

[33

Lin
—
0.0
——
Lon
PgE

;
it is generally counted positive in the counterclockwise sense. Because of conformal
pr operty, the geodetic azimuth of the geodesic is preserved during the mapping and it
m ust be equal to the angle between the tangents on the mapped meridian and mapped
ge odesic as shown. The symbols T and t denote the grid azimuth of the mapped
ge odesic and the rectilinear chord, respectively. The small angle

γ

[33

= T

− t is

∆ t

t , the meridian

ca lled the arc-to-chord correction. It is related to the grid azimuth

, and the azimuth of the geodesic co nvergence γ α by

α − γ − ¯

∆ t = t (9.15)

Figure 9.7 Mapping elements.