Global Positioning System Reference
In-Depth Information
unit vectors
e
,
n
,
u
into the orthogonal matrix
⎡
⎤
−
sin
λ
−
sin
ϕ
cos
λ
cos
ϕ
cos
λ
=
enu
=
⎣
⎦
.
F
cos
λ
−
sin
ϕ
sin
λ
cos
ϕ
sin
λ
(8.36)
0
cos
ϕ
sin
ϕ
Example 8.1
Let
P
be given by the coordinates
(ϕ, λ,
h
)
in the WGS 84 system:
40
◦
07
04
595 51
,
ϕ
=
.
277
◦
01
10
221 76
,
λ
=
.
h
=
231
.
562 m
.
runs eastward, from 0
◦
to 360
◦
.
The longitude
λ
We seek the
(
X
,
Y
,
Z
)
coordinates of
P
. The result is achieved by the
M
-file
g2c
:
X
=
596,915.961 m
,
Y
=−
4,847,845.536 m
,
Z
=
4,088,158.163 m
.
The reverse problem—compute
(ϕ, λ,
h
)
from
(
X
,
Y
,
Z
)
—requires an iteration
for
ϕ
and
h
. Directly
λ
=
arctan
(
Y
/
X
)
. There is quick convergence for
h
N
ϕ
,
starting at
h
=
0:
arctan
−
1
Y
2
1
−
(
2
−
f
)
fN
ϕ
Z
√
X
2
ϕ
ϕ
=
,
from
h
(8.32):
N
ϕ
+
h
+
(8.37)
√
X
2
Y
2
+
h
from
ϕ
(8.30)-(8.31):
h
=
−
N
ϕ
.
(8.38)
cos
ϕ
For large
h
(or
ϕ
close to
π/
2) we recommend the procedure given in the
M
-file
c2gm
.
8.8
Universal Transverse Mercator Mapping
The geographical coordinates
locate a point on the reference ellipsoid. For
many practical purposes it is useful to have a coordinate representation in the two-
dimensional plane. The mapping of an ellipsoid into a plane may be done by a
conformal mapping
. Conformity leaves the shape of small figures while distances
must be scaled.
The most widely spread conformal mapping was introduced by the Department
of Defense in United States shortly after the Second World War. It is called the
(ϕ, λ)
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