Global Positioning System Reference
In-Depth Information
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There is, of course, no specification in conformal mapping as to the preservation of
the length of the geodesic. Typically, one is not explicitly interested in the length of
the projected geodesic
d
. The map
¯
s
, but needs the length of the rectilinear chord
−
d
, follows readily from the lengths of the geodesic on
the ellipsoid and the rectilinear chord. The factor
s
distance reduction,
∆
s
=
d
s
k
L
=
(9.16)
is called the line scale factor. It must not be confused with the point scale factor
k
.
See Equation (C.42).
The angle between rectilinear chords on the map at station
i
is
t
i,i
+
1
−
T
i,i
−
1
− ∆
t
i,i
−
1
+
δ
i
=
t
i,i
+
1
−
t
i,i
−
1
+
π =
T
i,i
+
1
− ∆
[33
2
2
π
(9.17)
The angle between the geodesics on either the ellipsoid or their mapped images is
Lin
—
-1.
——
Lon
PgE
π =
T
i,i
+
1
+ γ
i
−
T
i,i
−
1
+ γ
i
+
δ
i
=
π =
T
i,i
+
1
−
T
i,i
−
1
+
α
i,i
+
1
−
α
i,i
−
1
+
2
2
π
(9.18)
2
The difference
δ
i
− δ
i
= ∆
∆δ
i
≡
t
i,i
+
1
− ∆
t
i,i
−
1
(9.19)
is
the angular arc-to-chord reduction. Equations (9.17) to (9.19) do not depend on the
m
eridian convergence.
Even though the term map distortion has many definitions, one associates a small
[33
∆
s
with small distortions, meaning that the respective reductions in angle
an
d distance are small and perhaps even negligible. It is important to note that the
m
apping elements change in size and sign with the location of the line and with its
or
ientation. In order to keep
t
and
∆
s
small, we limit the area represented in a single
m
apping plane in size, thus the need for several mappings to cover large regions of
th
e globe. In addition, the mapping elements are also functions of elements specified
by
the designer of the map, e.g., the factor
k
0
, the location of the central meridian, or
th
e standard parallel.
∆
t
and
∆
9.2.2 The Angular Excess
Th
e angular reduction can be readily related to the ellipsoidal angular excess. The
su
m of the interior angles of a polygon of rectilinear chords on the map (see Figure
9.8) is
δ
i
=
180
◦
(n
−
2
)
×
(9.20)
i
The sum of the interior angles of the corresponding polygon on the ellipsoid, consist-
ing of geodesics, is
