Geoscience Reference
In-Depth Information
Mean Radius of Curvature
The value of the mean radius of curvature, R
A
, varies with direction, making the
surveying computations difficult. Therefore, in practice the ellipsoid within a
certain range will be considered as a sphere with an appropriate radius to meet
the desired precision based on practical problems. It is reasonable to take the
spherical radius as the mean radius of curvature R
A
(averaging over all directions).
The average of the radii of curvature of the normal sections in all directions at a
point on the ellipsoid is the mean radius of curvature at this point, denoted by R.
To facilitate derivations, rearrange (
5.30
) as follows:
N
N
R
A
ﾼ
e
0
2
cos
2
A cos
2
B
ﾼ
e
0
2
cos
2
A cos
2
B
1
þ
sin
2
A
þ
cos
2
A
þ
ð
5
:
33
Þ
MN
MN
N cos
2
A
cos
2
A
ﾼ
ﾼ
M sin
2
A
:
þ
e
0
2
cos
2
B
M sin
2
A
þ
M 1
þ
Since the variations in R
A
with A are symmetrical with respect to the meridian
and the prime vertical, one needs simply to find the average of R
A
in one quadrant.
The average of the continuous function y
ﾼ
f(x) on the closed interval [a, b] is:
ð
b
1
y
average
ﾼ
fx
ðÞ
dx
:
b
a
a
Hence, the average of R
A
is:
N
q
dA
p
MN
ð
2
ð
2
1
2
MN
N cos
2
A
2
ˀ
cos
2
A
R
ﾼ
M sin
2
A
dA
ﾼ
2
:
q
tan A
0
þ
0
0
1
þ
q
tan A; thus dt
N
q
dA
Set t
ﾼ
ﾼ
cos
2
A
. The above equation upon substitution
gives:
ð
1
p
MN
p
MN
2
ˀ
dt
R
ﾼ
t
2
ﾼ
:
ð
5
:
34
Þ
1
þ
0
This equation shows that the mean radius of curvature at a point on the ellipsoid
equals the geometric mean of the radii of curvature in the meridian and the prime
vertical at this point.
Substituting the expressions of M (
5.32
) and N (
5.31
) into (
5.34
) gives the
formula for computing R:
Search WWH ::
Custom Search