Global Positioning System Reference
In-Depth Information
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map
P
i+1
i
P
i-1
P
i
Figure 9.8
Angle on the map.
δ
i
=
180
◦
+
(n
−
2
)
×
ε
(9.21)
i
[33
where
ε
denotes the ellipsoidal angular excess. It follows from (9.19) to (9.21) that
Lin
—
0.0
——
Lon
PgE
ε
=
∆
t
i,i
+
1
−
∆
t
i,i
−
1
(9.22)
i
i
Th
e angular excess can therefore be computed from either the sum of interior geodesic
an
gles, Equation (9.22), or Expression (B.65), which uses the Gauss curvature.
9.2.3 Direct and Inverse Solutions on the Map
Ha
ving the grid azimuth
t
and the length of the rectilinear chord
d
, or the angle
δ
be
tween rectilinear chords, the rules of plane trigonometry apply in a straightforward
m
anner. In case the geodetic latitude and longitude are given, one can use the mapping
eq
uations to compute the map coordinates first. The direct and inverse solutions are
sh
own in Figures 9.9 and 9.10, respectively.
The accuracy of the expressions for the transverse Mercator mapping is sampled
in
Figure 9.11. First, the geodetic latitudes and longitudes
(ϕ
i
,
[33
λ
i
)
were mapped to
(x
i
,y
i
)
and then computed back to the ellipsoid giving
(ϕ
ic
,
λ
ic
)
, using the inverse
m
apping functions. The distance (9.3) is plotted. The figure shows the case
(
λ =
2°
,
0°
<ϕ
i
<
90°
)
. The discrepancies are symmetric with respect to longitude. In
th
e region
−
λ
2°
<
i
<
2° the discrepancies are even smaller. The maximum values
for
4° are about 0.4 mm and 3 mm, respectively.
The discrepancies seen in Figure 9.11 are the result of the combined contribution
of
truncation errors in the transverse Mercator mapping functions of Tables C.1 and
C.
2, as well as Expression (B.42) for the elliptic arc. The Lambert conformal mapping
λ =
3° or
λ =
P
1
(x
1
,y
1
)
λ
1
) , d
12
, t
12
⇔
P
1
(ϕ
1
,
↓
x
2
=
x
1
+
d
12
sin
¯
t
12
y
1
+
d
12
cos
t
12
y
2
=
Figure 9.9
Direct solution on the map.
