198 5 “Sphere to tangential plane”: polar (normal) aspect
Box 5.18 (Distortion energy density tr[C l G
−1
l ]/2 = (Λ
2
1 (∆)+Λ
2
2 (∆))/2 for various azimuthal map projections
of the sphere, normal aspect (polar aspect)).
Equidistant (Postel):
1
2
`
Λ
2
1 + Λ
2
2
´
=
1
2
sin
2 ∆ + ∆
2
sin
2 ∆
=
1
2
cos
2 Φ +
` π
2
− Φ
´ 2
cos 2 Φ
.
(5.114)
Conformal (UPS):
1
2
`
Λ
2
1 + Λ
2
2
´
=
1
cos 4 ∆
2
=
1
cos 4
` π
4
−
Φ
2
´ .
(5.115)
Equiareal (Lambert):
1
2
`
Λ
2
1 + Λ
2
2
´
=
1
2
1 + cos
4 ∆
2
cos 2 ∆
2
=
1
2
1 + cos
4
` π
4
−
Φ
2
´
cos 2
` π
4
−
Φ
2
´ .
(5.116)
Gnomonic:
1
2
`
Λ
2
1 + Λ
2
2
´
=
1
2
cos
2 ∆ + 1
cos 4 ∆
=
1
2
sin
2 Φ + 1
sin
4 Φ
.
(5.117)
Orthographic:
1
2
`
Λ
2
1 + Λ
2
2
´
=
1
2
`
1 + cos
2 ∆
´
=
1
2
`
1 + sin
2 Φ
´
.
(5.118)
Lagrange conformal:
1
2
`
Λ
2
1 + Λ
2
2
´
=
1
4
1
cos 4 ∆
2
=
1
4
1
cos 4
` π
4
−
Φ
2
´ .
(5.119)
Box 5.19 (Total surface element S or the area of a spherical cap).
S = R
2
Z 2π
0
dΛ
Z π/2
Φ
dΦ cos Φ = R
2
Z 2π
0
dΛ
Z ∆
0
d∆ sin ∆ ,
(5.120)
subject to
∆ :=
π
2
− Φ or Φ =
π
2
− ∆ ,
dΦ = −d∆ ,
Z π/2
Φ
dΦ = −
Z 0
∆
d∆ = +
Z ∆
0
d∆ ;
(5.121)
S = 2πR
2 ˆ
+ sin Φ
˜ π/2
Φ
= 2πR
2 (1 − sin Φ) ,
S = 2πR
2 ˆ − cos ∆
˜ ∆
0
= 2πR
2 (1 − cos ∆) .
(5.122)