Reference Ellipsoid and the Geodetic Coordinate System - Geodesy: Introduction to Geodetic Datum and Geodetic Systems - page 213

observed directions. This effect is termed the correction for deflection of the
vertical, denoted by ʴ 1 .

This correction serves to solve the problem of relations between spatial angles.
One effective means of solving such a problem is to create an auxiliary sphere and
to denote the spatial angles by the arc length of the auxiliary sphere; namely, to
reduce the spatial angles to angles on the spherical surface. Then, the problem can
be solved by employing the methods of solving spherical triangles.
As shown in Fig. 5.31 , we create an auxiliary sphere with the observation point
A as the center. Then we draw a normal passing through point A that intersects the
auxiliary sphere at Z, which is called the geodetic zenith; the plumb line passes
through A, which intersects the auxiliary sphere at Z 1 , which is called the astro-
nomical zenith. The deflection of the vertical

is the angle between the ellipsoidal
normal and the plumb line at the point. We draw a line through point A parallel to
the minor axis of the ellipsoid, intersecting the auxiliary sphere at P, known as the
north pole of the auxiliary sphere. ZP

ʼ

is the meridian on the auxiliary sphere,
which is the line of intersection between the ellipsoidal meridian plane and the
auxiliary sphere. M is the projection of the terrestrial target point m on the surface of
the ellipsoid. With the north pole and the meridian introduced, the azimuth in the
Am direction can be expressed on the auxiliary sphere since spatial angles do not
change by translating a straight line or plane. The deflection of the vertical

ʸ

on the
auxiliary sphere is a short arc length, so it can be decomposed into two perpendic-
ular components

ʼ

ʾ and ʷ , which are known as the components of the deflection of

the vertical ʼ in the meridian and in the prime vertical, respectively, as seen in

Fig. 5.31 :

ʾ ﾼ ʼ cos ʸ , ʷ ﾼ ʼ sin ʸ:

From Fig. 5.31 ,ifM lies in the vertical plane ZZ 1 O, whether the observed
directions are referred to the ellipsoidal normal or the plumb line, the vertical
plane is the same. In this instance, there will be no vertical deflection correction.
Hence, we refer to AO as the reference direction (zero direction of the circle). In
Fig. 5.31 , if measured with respect to the plumb line AZ 1 , the circle reading at the
target point m will be OR 1 ; measured with respect to the normal AZ, the circle
reading at the target point m will be OR. Hence, the effect of the deflection of the
vertical on the horizontal directions will be R 1 ).

R 1 MR is a right spherical triangle. By applying the law of sines, we have:

ʴ 1 ﾼ (R

sin R 1 R ð Þﾼ sin ð ʴ 1 Þﾼ sin 90 ð z 1 Þ sin q,

where z 1 is the observed zenith distance in the M direction. Applying the sine
theorem to the triangle MZZ 1 we get: