Global Positioning System Reference
In-Depth Information
Satellite
k
u
z
ρ
Receiver
i
α
n
e
FIGURE 8.10. Zenith distance
z
and azimuth
α
in the topocentric system
(
e
,
n
,
u
)
.
ordinates and geographical
(ϕ, λ,
h
)
coordinates:
X
=
(
N
ϕ
+
h
)
cos
ϕ
cos
λ,
(8.30)
Y
=
(
N
ϕ
+
h
)
cos
ϕ
sin
λ,
(8.31)
=
(
h
sin
2
N
Z
1
−
f
)
ϕ
+
ϕ.
(8.32)
+
b
2
Z
2
The
reference ellipsoid
is the surface given by
X
2
a
2
.The
radius of curvature in the prime vertical (which is the vertical plane normal to the
astronomical meridian) is given as
Y
2
+
=
a
N
ϕ
=
1
.
(8.33)
sin
2
−
f
(
2
−
f
)
ϕ
In spherical approximation, i.e.,
f
=
0, the unit vector
u
normal to the surface is
⎡
⎤
ϕ
λ
cos
cos
⎣
⎦
.
u
=
cos
ϕ
sin
λ
(8.34)
sin
ϕ
The tangent
n
and binormal
e
unit vectors are derivatives of
u
(
n
alludes to nor-
thing,
e
to easting, and
u
to up, i.e., the normal direction to the surface):
⎡
⎤
⎡
⎤
−
sin
ϕ
cos
λ
−
sin
λ
∂
u
∂ϕ
=
1
cos
∂
u
∂λ
=
⎣
⎦
⎣
⎦
.
n
=
−
sin
ϕ
sin
λ
and
e
=
cos
λ
ϕ
ϕ
cos
0
(8.35)
Verify that
e
u
.
The unit vectors
n
,
e
,
u
provide the natural coordinate frame at a point on the
reference ellipsoid; see Figure 8.10. For reasons of reference we collect the three
=
n
×
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