The geometry of the earth - Physical Geodesy - page 219

s 0 = Rψ =2 R sin − 1 l 0

2 R .

(5-112)

Ellipsoidal refinements of these formulas may be found in Rinner (1956).
As a matter of fact, spatial distances are independent of the vertical.
Therefore, the reduction formula (5-111) does not contain the deflection of
the vertical ε .

5.14 The astrogeodetic determination of the geoid

Helmert's formula
The shape of the geoid can be determined if the deflections of the vertical
are given. Helmert's formula

dN = − εds (5-113)

as given in (2-372) is the basic equation (Fig. 5.15). Integrating this relation,
we get

B

N B = N A − εds, (5-114)

A

where

ε = ξ cos α + η sin α (5-115)

is the component of the deflection of the vertical along the profile AB ,whose
azimuth is α (see Eq. (5-101)).
Formula (5-114) expresses the geoidal undulation as an integral of the
vertical deflections along a profile. Since N is a function of position, this
integral is independent of the form of the line that connects the points A and
B . This line need not necessarily be a geodesic on the ellipsoid, and α may in
the general case be variable. In practice, north-south profiles ( ε = ξ )oreast-
west profiles ( ε = η ) are often used. The integral (5-114) is to be evaluated

plumb line ellipsoid normal

"

geoid

"

dN

ds

ds

ellipsoid

s

Fig. 5.15. Relation between geoidal undulation and deflection of
the vertical