Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 225

Table 5.8 Computation for
reduction of the slope
distance (GRS80 Ellipsoid)

Elements Example 1 Example 2

D 5,432.321 m 9,876.543 m

H 1 826.93 m 4,254.23 m

H 2 837.65 m 4,876.47 m

36 42 0 32 12 0

B

63 47 0 120 24 0 A

S 5,431.600 m 9,849.871 m

D 0 R A
R A þ

D 3
24R A 2 þ

10 16 H m D 2 sin 2Bcos A, S ﾼ

H m þ

1 : 25 ð 5 : 64 Þ

where

q

D 2

1
2 H 1 þ

c
1

D 0

2

ﾼ ð H 2 H 1 Þ , H m ﾼ ð H 2 Þ , N ﾼ p ,

e 0 2 cos 2 B

þ

N

R A ﾼ

e 0 2 cos 2 B cos 2 A ,

1 þ

D is the known slope distance, accurate to 0.001 m; H 1 and H 2 are the ellipsoidal
heights at the two end points of the observed distance, accurate to 0.001 m; B is the
geodetic latitude of the origin point of the observed distance, accurate to zero
decimal place; A is the geodetic azimuth of the observed distance, accurate to
zero decimal place; and S is the geodesic distance on the ellipsoid that the slope
distance being reduced to, accurate to 0.001 m. For an example of computation see
Table 5.8 .

5.5.5 Relationship Between Astronomical Longitude
and Latitude and Geodetic Longitude and Latitude
(Formula for Deflection of the Vertical)

The two components of the deflection of the vertical, , are the two required

quantities for reducing the terrestrial observation elements to the ellipsoid.
According to the definition of astronomical longitude and latitude,

ʾ and ʷ

deter-
mine the direction of the plumb line at a given point while L and B determine the
direction of the ellipsoidal normal at this point. Thus,

ʻ and ˆ

ʾ and ʷ can be defined by the

four parameters , L, and B.

Given the fact that the geodetic latitude B is the angle between the equatorial
plane and the line that is normal to the reference ellipsoid, the angle between the
ellipsoidal normal and the minor axis of the ellipsoid is 90

ʻ , ˆ

B; namely, in

90 90 ˆ Fig. 5.31 , PZ .PZis the geodetic meridian

plane and PZ 1 is the astronomical meridian plane; hence, the angle between the two
meridian planes is

ﾼ B. Similarly, PZ 1 ﾼ

L. It is also known that PZ 0 ﾼ 90

ʻ B ʾ . For all the

quantities above, see Fig. 5.38 .