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Fig. 5.28 The spherical triangle (a) and plane triangle (b) whose corresponding sides are equal in
length
5.5 Relationship Between Terrestrial Elements
of Triangulateration and the Corresponding Ellipsoidal
Elements
5.5.1 Significance of and Requirements for Reduction
of Terrestrial Triangulateration Elements
to the Ellipsoid
Conventional geodesy determines the horizontal coordinates L and B of an Earth's
surface point and the height of a point above the Earth's surface as two separate
issues. In order to find L and B of a surface point, one needs to project the geodetic
control network actually established on the physical surface of the Earth to the
reference ellipsoid adopted, i.e., to reduce the geodetic observations to the reference
ellipsoid (see, e.g., Torge and M¨ ller 2012). That is to say, the sides measured on
the physical surface of the Earth between points must be reduced to the geodesic arc
length on the surface of the reference ellipsoid and the observed values of horizon-
tal directions and astronomical azimuths should be reduced to the geodesic direc-
tions and geodetic azimuths (Fig. 5.29 ). Then, all the calculations concerning the
geodetic control network will be carried out on the surface of the ellipsoid as a
two-dimensional problem. After adjustment computations, one will find the longi-
tude L and latitude B, which can be transformed into the plane coordinates x, y by
applying the specified mathematical relations. Adjustment computations of the
large-scale astro-geodetic network are usually completed as such.
For small-scale geodetic control networks, it will be inconvenient to carry out
adjustment computations on the ellipsoid since the mathematical properties of the
ellipsoid is far more complicated than that of the plane. Hence, again we can project
the elements of the geodetic control network on the surface of the ellipsoid onto the
plane and then carry out computations on the planar surface, (see Chap. 6).
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