Geography Reference
In-Depth Information
Now, we are well-prepared to solve the forward and backward transformation problems, which are
also called the
direct
and the
inverse transformations
, which can be characterized as follows. (i)
Direct transformation: given the longitude
λ
and the latitude
φ
of point
x
∈
S
r
in the conventional
equatorial frame of reference as well as the spherical coordinates
of the meta-North
Pole, find the meta-longitude
α
and the meta-latitude
β
(alternatively, the meta-colatitude
ψ
)
of an identical point in the meta-equatorial (oblique) frame of reference of the sphere
{
λ
0
,φ
0
}
2
r
. (ii)
Inverse transformation: given the meta-longitude
α
and the meta-latitude
β
in the meta-equatorial
(oblique) frame of reference as well as the spherical coordinates
S
of the meta-North Pole,
find the longitude
λ
and the latitude
φ
of an identical point in the conventional equatorial frame
of reference of the sphere
{
λ
0
,φ
0
}
2
S
r
.
S
r
equatorial as well as meta-equatorial (oblique) frame of reference
Fig. 3.7.
Horizontal section of
Solution (the third problem, direct transformation).
Such a problem can be immediately solved as outlined in Boxes
3.3
-
3.6
.First,wehaveparam-
eterized the transformation of reference frames
{
e
1
0
,
e
2
0
,
e
3
0
|O} → {
e
1
,
e
2
,
e
3
|O}
by means
of the pole position
{λ
0
,φ
0
}
. Second, the placement vector
x
∈
S
r
of a point of the reference
sphere is represented in both the conventional equatorial frame of reference
{
e
1
,
e
2
,
e
3
|O}
and
in the meta-equatorial (oblique) frame of reference
{
e
1
0
,
e
2
0
,
e
3
0
|O}
at the origin
O
.Third,
we substitute
by means of the backward trans-
formation of reference frames, our first setup. The final representation of the placement vector
x
(
λ, φ
;
λ
0
,φ
0
) is achieved in terms of (i) conventional equatorial coordinates
{
e
1
,
e
2
,
e
3
|O}
in favor of
{
e
1
0
,
e
2
0
,
e
3
0
|O}
{
λ, φ
}
and (ii)
of the meta-North Pole. The corresponding two coordinate trans-
formations
α
(
λ, φ
;
λ
0
,φ
0
)and
β
(
λ, φ
;
λ
0
,φ
0
) are derived in Box
3.5
. The three identities for
(i) cos
α
, (ii) sin
α
, and (iii) sin
β
or cos
ψ
are derived by representing (i)
x
0
=
r
cos
β
cos
α
, (ii)
y
0
=
r
cos
β
sin
α
, and (iii)
z
0
=
r
sin
β
=
r
cos
ψ
in the oblique frame of reference. The first
identity is also called
spherical sine lemma
, the second identity is also called
spherical sine-cosine
lemma
, and the third identity is called
spherical side cosine lemma
. Indeed, we have derived the
collective formulae of Spherical Trigonometry, however, in a way to be used for other reference
equatorial coordinates
{
λ
0
,φ
0
}
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