Geoscience Reference
In-Depth Information
Rearranging gives:
1
l
2
6
cos
2
B 1
l
2
6
cos
2
B 1
y
N
y
N
t
2
2
t
2
2
l cos B
ﾼ
1
þ
þ ʷ
ﾼ
1
þ ʷ
:
We replace l cos B on the right-hand side of the above equation with
N
, which
gives:
,
l
00
ρ
00
y
3
6N
3
y
N
t
2
2
cos B
ﾼ
1
þ ʷ
l
00
2
ρ
00
2
cos
2
B
,
y
2
N
2
y
4
3N
4
t
2
2
ﾼ
1
þ ʷ
and
l
00
4
ρ
00
4
cos
4
B
y
4
N
4
:
ﾼ
Substituting into (
6.64
), one obtains:
1
y
2
N
2
y
4
3N
4
þ
y
4
24N
4
:
1
2
t
2
2
2
4t
2
m
ﾼ
1
þ
1
þ ʷ
þ ʷ
5
It follows that:
y
2
2N
2
þ
y
4
24N
4
:
2
m
ﾼ
1
þ
1
þ ʷ
ð
6
:
65
Þ
V
2
N
2
1
R
2
1
2
, substituting into the above equation and replacing
N
4
with R
4
, we get the formula for the scale factor expressed by the Gauss plane
coordinates:
With
ﾼ
ﾼ
N
2
1
ð
þ ʷ
Þ
y
2
2R
2
þ
y
4
24R
4
:
m
ﾼ
1
þ
ð
6
:
66
Þ
Table
6.5
provides some approximate values for the scale factor.
Distortion Properties
Subtracting 1 from the point scale factor, i.e., (m
1) results in the distortion of
distance at this point. Equation (
6.66
) shows that the point scale factor m depends
only on position, not on direction (which agrees with the conditions for the
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