The geometry of the earth - Physical Geodesy - page 204

We may use a rectangular coordinate system; then the three coordinates
to be determined will be X, Y, Z . The parameters defining the direction of
the plumb line are conveniently taken to be Φ and Λ, astronomical latitude
and longitude. We can express the astronomical azimuth A , the measured
zenith angle z , and the spatial distance s in terms of these five parameters.
This will be the scope of Sect. 5.9.
This information is purely “geometric”. We need the terrestrial measure-
ments (especially Φ , Λ ,A ) in order to link this geometry to the gravity field
as represented by the plumb lines. The Bruns polyhedron is the best way to
show this geometrically.
Today, GPS is the best way to determine global rectangular coordinates
X, Y, Z or ellipsoidal coordinates ϕ, λ, h directly.

5.9 Global coordinates and local level coordinates

We shall use a Cartesian coordinate system XY Z introduced in Sect. 5.6.1,
global but not necessarily geocentric. The coordinates X, Y, Z form a vector
X . Thus, the vectors X i and X j represent two terrestrial points P i and
P j . We define the vector between these two points in the global coordinate
system by X ij = X j − X i .
In addition, we introduce a “local level system” referred to the tangential
plane to the level surface at a point P i and to the local vertical, which is
the tangent at P i to the natural plumb line defined by the astronomical
coordinates Φ and Λ, see Sect. 2.4. The axes n i , e i , u i of this local (tangent
plane) coordinate system at P i corresponding to the north, east, and up

Z

u i

n i

e i

P i

X i

Y

i i

X

Fig. 5.11. Global and local level coordinates