1-10 Two examples: Mercator Projection and Stereographic Projection 57
Box 1.25 (Left Korn–Lichtenstein equations, UMP of E
2
A 1 ,A 1 ,A 2 , harmonicity, orientation).
Left Korn–Lichtenstein equations:
(1st) P Λ = +
r
G 11
G 22
Q Φ , P Φ = −
r
G 22
G 11
Q Λ (2nd) ,
(1st) Q Λ = −
r
G 11
G 22
P Φ , Q Φ = +
r
G 22
G 11
P Λ (2nd) ;
(1.196)
r
G 11
G 22
= cos Φ
1 − E
2 sin
2 Φ
1 − E 2
⇔
r
G 22
G 11
=
1
cos Φ
1 − E
2
1 − E 2 sin
2 Φ
;
(1.197)
(UMP left) Q Φ =
A 1 (1 − E
2 )
1 − E 2 sin
2 Φ
1
cos Φ
, P Λ = A 1 (KL 2nd) ,
(UMP left) Q Λ = 0 , P Φ = 0 (KL 1st) .
(1.198)
Left integrability conditions:
∆ Λ,Φ P =
r
G 11
G 22
P Φ
!
Φ
+
r
G 22
G 11
P Λ
!
Λ
= 0 ,
∆ Λ,Φ Q =
r
G 11
G 22
Q Φ
!
Φ
+
r
G 22
G 11
Q Λ
!
Λ
= 0 ;
(1.199)
(1st)
r
G 11
G 22
P Φ = 0 ,
r
G 22
G 11
P Λ =
A 1 (1 − E
2 )
1 − E 2 sin
2 Φ
1
cos Φ
⇒
∆ Λ,Φ P =
r
G 11
G 22
P Φ
!
Φ
+
r
G 22
G 11
P Λ
!
Λ
=
„
A 1 (1 − E
2 )
1 − E 2 sin
2 Φ
1
cos Φ
«
Λ
= 0
(1.200)
q. e. d.
(2nd)
r
G 11
G 22
Q Φ = A 1 ,
r
G 22
G 11
Q Λ = 0
⇒
∆ Λ,Φ Q =
r
G 11
G 22
Q Φ
!
Φ
+
r
G 22
G 11
Q Λ
!
Λ
= 0
(1.201)
q. e. d.
Left orientation:
˛
˛
˛
˛
˛
P Λ P Φ
Q Λ Q Φ
˛
˛
˛
˛
˛
= P Λ Q Φ − P Φ Q Λ = A
2
1
1 − E
2
1 − E 2 sin
2 Φ
1
cos Φ
> 0
(1.202)
due to
−π/2 < Φ < +π/2 ⇒ cos Φ > 0
(1.203)
q. e. d.