Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 215

can range from a few seconds to over 10 00 . The vertical angle is very
close to 0 in the first- and second-order triangulation. In plain areas, the vertical
angle is usually about

In general, ʼ

30 0 , and in mountainous areas, it can presumably amount to

3 . Hence, the value of

ʴ 1 is usually a few tenths of a second. Corrections for
deflection of the vertical need to be applied to the first- and second-order triangu-
lations. If the deflection of the vertical and the vertical angle are both quite large,
correction for deflection of the vertical should be taken into consideration even in
the third- or fourth-order triangulations.
On the following occasions, correction for deflection of the vertical equals zero:

1. The plumb line coincides with the ellipsoidal normal, i.e., ʼ ﾼ 0, then ʴ 1 ﾼ 0

2. The target point is in the plane ZZ 1 O, i.e., A ﾼ ʸ , then ʴ 1 ﾼ 0

90 o , then 3. The target point is in the horizontal plane, i.e., Z 1 ﾼ ʴ 1 ﾼ 0

Correction for Skew Normals

After the correction ʴ 1 is applied, the direction value will be the direction of the

normal section Ab 0 in Fig. 5.32 . Here, the height of the observation point A has no
effect on the value of horizontal directions. To simplify, we set A on the ellipsoid. In
accordance with the requirements for reductions, the projection of the target point
B on the surface of the ellipsoid should be b rather than b 0 . Hence, the angular
difference between these two normal sections Ab 0 and Ab is ʴ 2, known as the skew

normal correction. Obviously, this correction is due entirely to the effect of the
height of the target point B on the reduced direction value.
In Fig. 5.32 , the ellipsoidal triangle Abb 0 is considered a plane triangle. By
applying the sine theorem, it gives:

sin A 1
S

bb 0

00

ʴ 2 ﾼ ˁ :

This serves to show that bb 0 needs to be computed first in order to obtain

ʴ 2 .In

the triangles Bbb 0 and BRK a, we see that:

bb 0 ﾼ H 2 ʸ ,

K a R
BR

K a R
N 2 :

ʸ ﾼ

The “2” denotes the corresponding value at point B and the subscript “1” in the
following equations denotes the corresponding value at point A. From Fig. 5.32 :

K a R ﾼ K a K b cos B 2 ,

and