Geography Reference
In-Depth Information
3
,
I
3
}
;
X
(
Λ, Φ, R
)=
E
1
R
cos
Φ
cos
Λ
+
E
2
R
cos
Φ
sin
Λ
+
E
3
R
sin
Φ ∈{
R
{X,Y,Z}→{Λ, Φ, R}
:
(5.3)
X
+ 180
◦
2
sgn
Y
sgn
X
+1
,
Λ
(
X
) = arctan
Y
1
2
sgn
Y
1
−
−
Z
√
X
2
+
Y
2
,
Φ
(
X
) = arctan
R
=
√
X
2
+
Y
2
+
Z
2
.
Matrices F, G, H, J, K, and I (elements :) (
a, b, c, d
;
e, f, g
;
l,m,n
):
F=
ab
=
=
1
√
G
11
0
1
R
cos
Φ
0
0
2
×
2
,
∈
R
(5.4)
1
R
1
√
G
22
cd
0
G
=
ef
=
R
2
cos
2
Φ
0
R
2
2
×
2
,
∈
R
fg
0
H=
lm
=
−
R
cos
2
Φ
0
0
2
×
2
,
∈
R
(5.5)
mn
−R
⎡
⎤
2
,
K=
R
0
−R
cos
Φ
sin
Λ −R
sin
Φ
cos
Λ
+
R
cos
Φ
cos
Λ −R
sin
Φ
sin
Λ
0
⎣
⎦
∈
R
3
×
2
×
2
,
J=
∈
R
(5.6)
0
R
R
cos
Φ
R
2
,
I=I
2
=
10
1
R
,k
=
1
2
×
2
.
h
=
−
∈
R
(5.7)
01
Christoffel symbols:
1
11
=
1
=
2
=
2
=0
,
1
=
−
tan
Φ,
22
12
22
12
2
11
=sin
Φ
cos
Φ
=
1
2
sin 2
Φ.
(5.8)
5-1 General Mapping Equations
Setting up general equations of the mapping “sphere to plane”: the azimuthal projection in
the normal aspect (polar aspect).
There are two basic postulates which govern the setup of general equations of mapping the sphere
S
2
2
2
R
of radius
R
to a tangential plane
T
X
0
S
R
, which is attached to a point
X
0
∈
T
S
R
.Letthe
tangential plane be covered by polar coordinates
{
α,r
}
. Then these postulates read as follows.
Search WWH ::
Custom Search