The geometry of the earth - Physical Geodesy - page 188

the known station A to achieve the high accuracy. Note that the resulting
coordinates are obtained in the WGS 84.

This concludes the short introduction how the user of GPS gets WGS 84
coordinates, i.e., geocentric rectangular coordinates X, Y, Z or, computed
from them, ellipsoidal coordinates ϕ, λ, h ; see Sect. 5.6.1.

5.5 Projection onto the ellipsoid

Let us establish the position of a point P by means of the natural coor-
dinates Φ , Λ ,H . Then we may project it onto the geoid along the (slightly
curved) plumb line. The orthometric height is the distance between P and its
projection P 0 onto the geoid, measured along the plumb line (Fig. 5.4). Al-
though this mode of projection is entirely natural, the geoid is not suited for
performing computations on it directly; the point P 0 is, therefore, projected
onto the reference ellipsoid by means of the straight ellipsoidal normal, thus
getting a point Q 0 on the ellipsoid. In this way, the earth's surface point P
and the corresponding point Q 0 on the ellipsoid are connected by a double
projection, that is, by two projections which are performed one after the
other and which are quite analogous, the orthometric height H = PP 0 cor-
responding to the geoidal undulation N = P 0 Q 0 . This double projection is
called Pizzetti's projection .
It is much simpler to project the point P from the physical surface of the
earth directly onto the ellipsoid through the straight ellipsoidal normal, thus
obtaining a point Q . The distance PQ = h is the ellipsoidal height, i.e., the
height above the ellipsoid. The earth's surface point P is then determined
by the ellipsoidal height h and the ellipsoidal coordinates ϕ, λ of Q on the
ellipsoid so that the ellipsoidal coordinates ϕ, λ, h take the place of the natural
coordinates Φ , Λ ,H .Thisiscalled Helmert's projection .

P

earth's

"
h H ±

surface

P 0

geoid

N

ellipsoid

Q Q 0

Fig. 5.4. The projection of Helmert and of Pizzetti