Geography Reference
In-Depth Information
Box 3.12 (From the equatorial frame of reference
{
E
1
,
E
2
,
E
3
|O}
to the meta-equatorial
(oblique) frame of reference
{
E
1
,
E
2
,
E
3
|O}
: the direct transformation).
(i) The first identity:
X
=
R
cos
B
cos
A
⇒
X
R
cos
B
,
cos
A
=
(3.72)
1
cos
B
(cos
Φ
cos
Λ
cos
Ω
+cos
Φ
sin
Λ
sin
Ω
)
,
cos
A
=
cos
Φ
cos
A
=
cos
B
cos(
Λ − Ω
)
.
(ii) The second identity:
Y
=
R
cos
B
sin
A
⇒
Y
R
cos
B
,
sin
A
=
(3.73)
1
cos
B
(
−
cos
Φ
cos
Λ
sin
Ω
cos
I
+cos
Φ
sin
Λ
cos
Ω
cos
I
+sin
Φ
sin
I
)
,
sin
A
=
sin
A
=
1
cos
B
(cos
Φ
cos
I
sin(
Λ − Ω
)+sin
Φ
sin
I
)
.
(iii) The third identity:
Z
=
R
sin
B
⇒
sin
B
=
Z
R
,
(3.74)
sin
B
=cos
Φ
cos
Λ
sin
Ω
sin
I
−
cos
Φ
sin
Λ
cos
Ω
sin
I
+sin
Φ
cos
I,
sin
B
=
−
cos
Φ
sin
I
sin(
Λ
−
Ω
)+sin
Φ
cos
I.
Box 3.13 (The forward problem of transforming spherical frames of reference. Input variables:
Λ, Φ, Ω, I
. Output variables:
A, B
).
(i) The first and second identities:
tan
A
=
cos
I
sin(
Λ
Ω
)+tan
Φ
sin
I
cos(
Λ
−
.
(3.75)
−
Ω
)
Alternatives :
sin
A
=
cos
Φ
cos
B
(cos
I
sin(
Λ
−
Ω
)+tan
Φ
sin
I
)
,
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