Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 222

Fig. 5.35 Derivations of ε

We take out the spherical triangle ZZ 1 M in Fig. 5.31 (see Fig. 5.35 ) and draw a
line Z 1 Z 00 perpendicular to ZM. Since q is a small quantity, ZZ 00 ﾼ ε . Also in

ZZ 1 Z 00 , we see that: ʔ

ε ﾼ u cos(A ʸ )

ﾼ u cos ʸ cos A þ u sin ʸ sin A

ﾼ ʾ cos A þ ʷ sin A

Hence:

z ﾼ z 1 þ ʾ cos A þ ʷ sin A : ð 5 : 60 Þ

are
the components of deflection of the vertical in the meridian and in the prime vertical
at the observation point, respectively. A denotes the geodetic azimuth of the
observed direction.

This is the reduction formula of the observed zenith distance, where ʾ . and ʷ

5.5.4 Reduction of the Observed Slope Distance
to the Ellipsoid

The length observed using the rangefinder is known as the slope distance between
two points on the Earth's surface. Reducing the slope distance to the ellipsoid,
namely converting the slope distance D to the geodesic distance S (Fig. 5.36 ), is
termed the reduction of the slope distance.
We will now derive the formula for the reduction of the slope distance over short
distances. Two approximations are made: first, K a and K b are considered to coin-
cide; and second, geodesic S is considered an arc of a great circle. Hence, the
reduction in Fig. 5.36 becomes finding the solution of the plane triangle in Fig. 5.37 .