Global Positioning System Reference
In-Depth Information
FIGURE 2.6
A basic ellipse with accessory lines.
The ellipticity
e
p
is defined as
a
e
−
b
e
e
p
=
(
2
.
23
)
a
e
where
a
e
=
6378137
±
2m,
b
e
=
6356752
.
3142 m,
e
e
=
0
.
0818191908426, and
e
p
=
0
.
00335281066474.
(
6
,
7
)
The value of
b
e
is calculated from
a
e
; thus, the
result has more decimal points.
From the user position
P
draw a line perpendicular to the ellipse that intercepts
it at
A
and the
x
-axis at
C
. To help illustrate the following relation a circle with
radius equal to the semi-major axis
a
e
is drawn as shown in Figure 2.6. A line
is drawn from point
A
perpendicular to the
x
-axis and intercepts it at
E
and the
circle at
D
. The position
A
(
x
,
y
) can be found as
x
=
OE
=
OD
cos
β
=
a
e
cos
β
DE
b
e
(a
e
sin
β)
b
e
z
=
AE
=
a
e
=
a
e
=
b
e
sin
β
(2.24)
The second equation can be obtained easily from the equation of a circle
x
2
+
z
2
a
e
and Equation (2.21). The tangent to the ellipse at
A
is
dz/dx
. Since line
CP
is perpendicular to the tangent,
=
dx
dz
tan
L
=−
(
2
.
25
)
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