1-10 Two examples: Mercator Projection and Stereographic Projection 67
Box 1.34 (Conformally flat left and right manifold, ellipsoid-of-revolution versus sphere).
Left manifold,
{Λ, Φ} left coordinates:
Right manifold,
{λ, φ} right coordinates:
dS
2 =
A 1 cos
2 Φ
1 − E 2 sin
2 Φ
dΛ
2 +
A
2
1 (1 − E
2 )
2
(1 − E 2 sin
2 Φ) 3 dΦ
2 .
ds
2 = r
2 cos
2 φ dλ
2 + r
2 dφ
2 .
(1.244)
Left Cauchy–Green matrix:
Right Cauchy–Green matrix:
dP
2 + dQ
2 =
`
P
2
Λ + Q
2
Λ
´
dΛ
2 +
d p
2 + dq
2 =
`
p
2
λ + q
2
λ
´
dλ
2 +
+2 (P Λ P Φ + Q Λ Q Φ ) dΛdΦ+
+ 2 ( p λ p φ + q λ q φ ) dλdφ+
+
`
P
2
Φ + Q
2
Φ
´
dΦ
2 =
+
`
p
2
φ + q
2
φ
´
dφ
2 =
= f
2 (Φ)dΛ
2 + f
2 (Φ)dΦ
2
= g
2 (φ)dλ
2 + g
2 (φ)dφ
2
⇔
⇔
l c 11 = f
2 (Φ) , l c 12 = 0 , l c 22 = f
2 (Φ) ;
r c 11 = g
2 (φ) , r c 12 = 0 , r c 22 = g
2 (Φ) ;
(1.245)
˛
˛ C l − Λ
2
l G l
˛
˛ = 0
˛
˛ C r − Λ
2
r G r
˛
˛ = 0
⇔
⇔
˛
˛
˛
˛
˛
f
2 (Φ) − Λ
2
l G 11
0
0
f
2 (Φ) − Λ
2
l G 22
˛
˛
˛
˛
˛
= 0 ;
˛
˛
˛
˛
˛
g
2 (φ) − Λ
2
r g 11
0
0
g
2 (φ) − Λ
2
r g 22
˛
˛
˛
˛
˛
= 0 ;
(1.246)
l Λ
2
1 =
f
2 (Φ)
G 11
=
f
2 (Φ)
A
2
1 cos 2 Φ
(1 − E
2 sin
2 Φ) ,
r Λ
2
1 =
g
2 (φ)
g 11
=
g
2 (φ)
r 2 cos 2 φ
,
l Λ
2
2 =
f
2 (Φ)
G 22
=
r Λ
2
2 =
g
2 (φ)
g 22
=
=
f
2 (Φ)
A
2
1 (1 − E 2 )
(1 − E
2 sin
2 Φ) ,
=
g
2 (φ)
r 2 ,
f
2 (Φ) =
(1 − E
2 )
2
(1 − E 2 sin
2 Φ) 2
f
2 (Φ)
cos 2 Φ
g
2 (φ) =
g
2 (φ)
cos 2 φ
⇒
⇒
l Λ
2
1 = l Λ
2
2 = Λ
2
l ;
r Λ
2
1 = r Λ
2
2 = Λ
2
r ;
(1.247)
r
2 cos φ
g 2 (φ)
=
r
2 cos
2 φ
4r 2 tan
2
` π
4
−
φ
2
´ =
1
4
sin
2
` π
2
− φ
´
tan
2
` π
4
−
φ
2
´ =
sin
2
` π
4
−
φ
2
´
cos
2
` π
4
−
φ
2
´
tan
2
` π
4
−
φ
2
´
=
= cos
4
` π
4
−
φ
2
´
;
(1.248)
Λ
2
l =
1 − E
2 sin
2 Φ
A
2
1 cos 2 Φ
f
2 (Φ)
Λ
2
r =
1
cos 4
` π
4
−
φ
2
´
⇔
⇔
λ
2
l =
A
2
1 cos
2 Φ
1 − E 2 sin
2 Φ
1
f 2 (Φ)
.
λ
2
r = cos
4
„
π
4
−
φ
2
«
.
(1.249)
Conformally flat left manifold:
Conformally flat right manifold:
dS
2 = λ
2
l
`
dP
2 + dQ
2 ´
.
ds
2 = λ
2
r
`
dp
2 + dq
2 ´
.
(1.250)