318 15 “Ellipsoid-of-revolution to cylinder”: transverse aspect
Example 15.2 (Universal Polar Stereographic Projection (UPS)).
x =
2A 1
√
1 − E 2
1 − E
1 + E
E/2
tan
π
4
−
B
2
1 + E sin B
1 − E sin B
E/2
cos L ,
y =
2A 1
√
1 − E 2
1 − E
1 + E
E/2
tan
π
4
−
B
2
1 + E sin B
1 − E sin B
E/2
sin L .
(15.17)
The matrix of the metric of the ellipsoid-of-revolution E
2
A 1 ,A 1 ,A 2
is represented by
G MN =
G 11 G 12
G 12 G 22
=
⎡
⎣
A
2
1 cos
2 B
1−E 2 sin 2 B
0
0
A
2
1 (1−E
2 )
2
(1−E 2 sin 2 B) 3
⎤
⎦ .
(15.18)
The mapping equations of type UPS imply
x L = −f (B) sin L , x B = f
(B) cos L ,
y L = f (B) cos L , y B = f
(B) sin L ,
(15.19)
subject to
f (B) :=
:=
2A 1
√
1 − E 2
1 − E
1 + E
E/2
tan
π
4
−
B
2
1 + E sin B
1 − E sin B
E/2
,
f
(B) :=
:= −
2A 1
√
1 − E 2
1 − E
1 + E
E/2 tan
π
4 −
B
2
cos B
1 − E
2
1 − E 2 sin
2 B
1 + E sin B
1 − E sin B
E/2
=
= −
1 − E
2
cos B(1 − E 2 sin
2 B)
f (B) .
(15.20)
Korn–Lichtenstein equations:
x L =
G 11
G 22
y B , x B = −
G 22
G 11
y L , y L = −
G 11
G 22
x B , y B =
G 22
G 11
x L ,
(15.21)
G 11
G 22
= cos B
1 − E
2 sin
2 B
1 − E 2
⇒
y B = −
1 − E
2
cos B(1 − E 2 sin
2 B)
f (B) sin L = f
(B) sin L ,
y L = −
cos B(1 − E
2 sin
2 B)
1 − E 2
f
(B) cos L = f (B) cos L .
(15.22)
End of Example.