Geoscience Reference
In-Depth Information
a
p
e
2
1
c
V
2
:
R
ﾼ
ﾼ
ð5:35Þ
W
2
M, N, and R at a point on the ellipsoid are all measured inward along the normal
at this point to the surface. Their lengths are generally different from each other.
Comparing (
5.31
), (
5.32
), and (
5.35
), one can obtain their relations as:
< 90
o
N
>
R
>
M 0
ð
B
Þ :
When the point is at the poles, we have:
90
o
N
90
ﾼ
R
90
ﾼ
M
90
ﾼ
cB
ð
ﾼ
Þ:
Hence, c is the radius of curvature at the poles.
5.4.3 Length of a Meridian Arc and Length of a Parallel Arc
When geodetic computations are performed on the surface of the ellipsoid, such as
computation of the Gauss projection, the formulae for the length of a meridian arc
and the length of a parallel arc are needed. Derivations of the formulae are thereby
given next.
Formula for Length of a Meridian Arc
As shown in Fig.
5.14
, let P
1
and P
2
be two points along the meridian at latitudes B
1
and B
2
, respectively; compute the meridian arc length X between the two points P
1
and P
2.
If the meridian is a circular arc, the length of the arc is the product of the radian
measure of the central angle subtended by the arc times the radius of the circle.
However, the meridian is an elliptical arc. Its arc length must be calculated using
integration. Take a short arc segment on the meridian, i.e., the arc element (differ-
ential of arc) PP
0
ﾼ
dX. The central angle subtended by the arc (difference in
latitude) is denoted by dB. Let the point P be at the latitude B, and the latitude of
point P
0
will be B +dB. Let the meridional radius of curvature at the point P be M,
and arc element dX can be considered the circular arc of radius M; thus:
dX
ﾼ
MdB
:
ð
5
:
36
Þ
To compute the arc length X between P
1
and P
2
is to find the integral of dX
between B
1
and B
2
, namely:
Search WWH ::
Custom Search