Geography Reference
In-Depth Information
Note that all the partial derivatives that are quoted in these boxes can be computed by taking
advantage of the identities d tan
x
=(1+tan
2
x
)d
x
and d
x
=dtan
x/
(1 + tan
2
x
),whichleadto
the recursive scheme presented in Box
3.22
.
Box 3.22 (Recursive relations for the partial derivatives
l
ij
and
b
ij
up to order three,
x ∈
{Λ, Φ
).
(1 + tan
2
x
)
2
∂
tan
x
2
∂x
∂A
=+
∂
tan
x
∂A
∂
2
x
∂A
2
=+
2tan
x
1
1+tan
2
x
,
∂A
1
1+tan
2
x
∂
tan
x
∂A
+
,
(1 + tan
2
x
)
3
∂
tan
x
3
∂
3
x
∂A
3
=
2
1
−
3tan
x
−
∂A
(1 + tan
2
x
)
2
∂
tan
x
∂
2
tan
x
∂A
2
+
6tan
x
−
(3.142)
∂A
∂
3
tan
x
∂A
3
1
1+tan
2
x
+
,
(1 + tan
2
x
)
2
∂
tan
x
∂
tan
x
∂A
.
∂
2
x
∂A∂B
=+
∂
2
tan
x
∂A∂B
−
1
1+tan
2
x
2tan
x
∂B
3-46 Inverse Transformation of Oblique Quasi-Spherical
Longitude/Latitude
X
,Y
,Z
}
Let us depart from the representation (
3.108
) of oblique Cartesian coordinates
{
in
terms of oblique quasi-spherical longitude/latitude
. The inverse map relates these oblique
Cartesian coordinates to oblique quasi-spherical longitude/latitude:
{
A, B
}
tan
A
=
Y
Y
,
Z
√
X
2
+
Y
2
.
tan
B
=
(3.143)
As soon as we implement the transformation of normal Cartesian coordinates
{X, Y, Z}
into
oblique Cartesian coordinates
{X
,Y
,Z
}
of type (
3.104
)and(
3.105
)aswellasthesurface
normal ellipsoidal longitude/latitude
{Λ, Φ}
into normal Cartesian coordinates
{X, Y, Z}
of
type (
3.129
), we are led to
X
=
X
cos
Ω
+
Y
sin
Ω
=
A
1
1
=
cos
Φ
(cos
Λ
cos
Ω
+sin
Λ
sin
Ω
) =
(3.144)
E
2
sin
2
Φ
−
A
1
1
cos
Φ
cos(
Λ − Ω
)
,
=
E
2
sin
2
Φ
−
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