Geography Reference
In-Depth Information
3
,δ
IJ
}
is represented either in the
orthonormal triad
{
E
1
,
E
2
,
E
3
}
, which is oriented along the ordered principal axes of the ellipsoid-of-revolution,
or is represented in the
transverse orthonormal triad {
E
1
,
E
2
,
E
3
}
=
{−
E
1
,
E
3
,
E
2
}.
X
I
∈
X
I
∪
X
II
X
II
constitute the
minimal atlas
of
E
The placement vector
X
∈{
R
A
1
,A
2
.
Inverse formulae {L, B}
(
X, Y, Z
)and
{
(
X, Y, Z
)aregivenby
Grafarend and Lohse
(
1991
). Note that in both charts (local
coordinates
U, V
}
, respectively) the surface normal vector
G
3
is represented by
(
20.4
), motivating why the local coordinates
{
L, B
}
and
{
U, V
}
{
L, B
}
and
{
U, V
}
, respectively, are called
surface
normal coordinates
.
G
3
=+
E
1
cos
B
cos
L
+
E
2
cos
B
sin
L
+
E
3
sin
B
versus
(20.4)
G
3
=+
E
1
cos
V
cos
U
+
E
2
cos
V
sin
U
+
E
3
sin
V.
2
3
,δ
IJ
The embedding
E
A
1
,A
2
⊂{
R
}
is characterized by the
mapping equations
tan
L
=
Y
X
Z
versus tan
U
=
−
E
2
)
X
,
(20.5)
(1
−
Z
(1
−
E
2
)
Y
tan
B
=
−
E
2
)
√
X
2
+
Y
2
versus tan
V
=
(1
E
2
)
X
2
+
Z
2
.
(20.6)
(1
−
−
E
2
A
1
,A
2
. First chart: “surface normal” ellipsoidal longitude, ellipsoidal latitude
Fig. 20.2.
The minimal atlas of
{
0
<L<
2
π,
−
π/
2
<B<
+
π/
2
}
. Second chart: “surface normal” meta-longitude, meta-latitude
{
0
<U<
2
π,
. Half ellipse
L
=0,SouthPole
B
=
π/
2, North Pole
B
=+
π/
2 excluded in the first
chart. Half circle
U
= 0, meta-South Pole
V
=
−
π/
2
<V <
+
π/
2
}
−
π/
2, meta-North Pole
V
=+
π/
2 excluded in the second chart.
{
E
1
,E
2
}
span the equator plane,
{
E
1
,E
2
}
=
{−
E
1
,E
3
}
span the meta-equator plane
Search WWH ::
Custom Search