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OK
a
ﾼ
Q
1
K
a
sin B
1
Q
2
K
b
sin B
2
:
OK
b
ﾼ
Ne
2
,we
With the length of the normal on the lower side of the equator QK
ﾼ
have:
N
1
e
2
sin B
1
OK
a
ﾼ
:
N
2
e
2
sin B
2
OK
b
ﾼ
OK
b
,soK
a
and K
b
do not
coincide, which indicates that AK
a
and BK
b
do not lie in the same plane. When both
points A and B are situated on the same meridian or on the same parallel, the normal
section and reverse normal section will be coincident, which is a special case.
The above equations also show that when B
2
>
From the above equations, if B
1
6ﾼ
B
2
, then OK
a
6ﾼ
OK
a
. Fig-
ure
5.18
indicates that K
a
is above K
b
. The two normal section planes AK
a
B and
BK
b
A intersect at the chord AB. The normal sections AaB and BbA are created by
intersecting the planes containing the normals to the ellipsoid (normal section
planes) with the surface of the ellipsoid. BbA is on the upper side, while AaB is
on the lower side. It follows that the normal section from the point with higher
latitude to the point with lower latitude is on the upper side, whereas that from the
point with lower latitude to the point with higher latitude is on the lower side. We
term AaB the normal section of point A, while BbA is the reverse normal section of
A. According to the above law, we can draw the relationship of position between the
normal section and reverse normal section from point A to B in different quadrants,
as illustrated in Fig.
5.19
.
Reciprocal normal sections usually do not coincide, except for the two cases
where the two points are situated on the same meridian or at the same latitude. The
angle
B
1
, then OK
b
>
of the normal section and the reverse normal section between two points
(Fig.
5.20
) can be up to 0.004
00
when their distance apart S is 25 km or even a few
hundredths of a second when S
ʔ
ﾼ 50 km (directly proportional to the square of the
distance) in first-order triangulation. It must be taken into account in computation of
first-order triangulation.
The existence of reciprocal normal sections brings inconvenience for geodetic
computations. Let A, B, and C be three points on the surface of the ellipsoid. Their
longitudes are L
C
>
B
A
. In reciprocal triangular
observations, the case in Fig.
5.20
will occur where the three angles A, B, C formed
by the normal sections cannot constitute a triangle, which is to say, the reciprocal
normal sections have caused fracture of the geometric figure.
Obviously we cannot base our computations on a fractured figure but instead
choose a unique curve between the two points to replace the reciprocal normal
sections. There are many kinds of single curves between two points on the ellipsoid.
However, the curve between the two points should be unique and possess distinct
geometric properties (such as the shortest line between two points on the ellipsoid),
L
B
>
L
A
, and latitudes B
B
>
B
C
>
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