Geoscience Reference
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Table 5.2 The law of changes in the radius of curvature in the prime vertical (N) with latitude (B)
B
N
Note
8
<
:
a
c
1
B
ﾼ
0º
At the equator, the prime vertical coincides with the equator.
N is the equatorial radius of curvature
N
0
ﾼ
p
e
0
2
þ
0º
<
B
<
90º a
<
N
<
c
N increases with the increasing latitude, its value ranging
from a to c
8
<
:
a
1
B ﾼ 90º
At the poles, the prime vertical and the meridian coincide
with each other. N is the polar radius of curvature
p
e
2
N
90
ﾼ
c
and
5.13
, its trace is marked by a straight line OK
N
connecting the center of the
ellipsoid O on the minor axis with the point K
N
under it.
Radius of Curvature of the Meridian
0
o
into (
5.31
) gives:
Substituting A
ﾼ
N
N
V
2
,
R
0
ﾼ
M
ﾼ
e
02
cos
2
B
ﾼ
1
þ
W
2
And with (
5.31
) and V
2
ﾼ
e
2
, we obtain:
1
e
2
a 1
ð
Þ
c
V
3
:
M
ﾼ
ﾼ
ð
5
:
32
Þ
W
3
Similarly to N, M is also a function of B and increases with the increasing
absolute value of latitude B. The law of its changes is detailed in Table
5.3
.
Tables
5.2
and
5.3
show that c is the radius of curvature of the normal section at
the pole. On the surface of the ellipsoid, the normal sections that pass through the
poles are meridians. Hence, the polar radius of curvature c coincides with the
meridional radius of curvature at the poles.
Table
5.3
also indicates that when a point P on the ellipsoid moves from the
equator along the meridian to the North Pole, the endpoint K
0
of the meridional
radius of curvature also moves relative to it. As shown in Fig.
5.13
, its trajectory is
marked by an asteroid K'
0
K'K
N
close to the center of the ellipsoid.
In (
5.30
), when A
90
(or 270
), the value of R
A
is largest. Hence, N and M are the maximum and
minimum of R
A
, which means that N is always greater than M except at the poles.
When A starts from due North 0
and increases eastward to 90
, R
A
shifts
progressively from M to N; when A starts from due East 90
and increases eastward
to 180
, R
A
shifts gradually from N to M. Therefore the radius of curvature R
A
in any
0
(or 180
), the value of R
A
is smallest; when A
ﾼ
ﾼ
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