Global Positioning System Reference
In-Depth Information
P 1 ϕ 1 , 0 ,
λ 1 , 0 ,x 1 , 0 ,y 1 , 0 ,P 2 ϕ 2 , 0 ,
λ 2 , 0 ,x 2 , 0 ,y 2 , 0
1
2
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d 12 , 0 , t 12 , 0 ,
s 12 , 0
γ 1 , 0 ,k 1 , 0 ,k 2 , 0 ,k m, 0 ,
t 12 , 0 ,
Mapping reductions using functions for t and s .
Figure 9.13
The available functions (t geod ,s geod ) or (t map ,s map ) are typically again a result of
series expansions and truncations. The derivation of (9.29) and (9.30) requires at least
as much algebraic work as the derivation of the GML functions and the mapping
equations because the mapped geodesic is involved. These functions are, strictly
speaking, not needed if we use the GML functions as demonstrated above. The
reduction scheme using the explicit functions of
t and
s is shown in Figure 9.13. It
[33
does not contain the geodesic azimuth
α 12 , 0 explicitly. The point scale factor k serves
merely as an auxiliary quantity to express
s in a compact form. The subscripts of
k indicate the point of evaluation. In the case of m , k is evaluated at the midpoint
[ 1 +
Lin
0.7
——
Nor
PgE
λ 1 + λ 2 )/ 2]. See Table 9.5 for a listing of the expressions.
Computing the angular access of a polygon is a convenient way to verify the
accuracy of the
ϕ 2 )/ 2 ,(
t expressions. First, we compute the angular excess of a geodesic
polygon on the ellipsoid using the GML functions. For reasons of convenience, we
choose equally spaced points from a geodesic circle as the vertices of the geodesic
polygon (points on the geodesic circle and the center of the circle are connected
by a geodesic with the same length). The sides of the polygon must, of course, be
geodesics. The latter is automatically achieved since the geodesic angle between the
sides of the polygon is computed from the GML functions. The conformal mappings
are specified by the central meridian and the standard parallel that go through the
center of the circle, as well as taking k 0 =
[33
1. The radius of the geodesic circle, i.e.,
the size of the polygon, was varied from zero to the equivalent of about 2°. Figure
9.14 shows the differences in the angular excess for these polygons as computed from
the GML functions and the explicit functions of Table 9.5. The figure shows that the
TM and LC expressions in Table 9.5 are of the same accuracy within the region of
computation and that they agree to about 0.1 arcsec with the GML computation.
Since the angular excess is independent of the specific conformal mapping the lines
in Figure 9.14 coincide, in theory.
Explicit Functions for
t and
TABLE 9.5
s
(x 2 +
2 x 1 )(y 2
y 1 )
( 2 y 1 +
y 2 )(x 1
x 2 )
TM:
t 1 =
LC:
t 1 =
6 k 0 R 1
6 k 0 R 0
1
k 1 +
s
d =
1
k L
1
6
4
k m +
1
k 2
s ( 1
s
=
k L )
 
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