“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 224

Conformal (UPS):

1 1

cos 4 2

1

2 ( Λ 1 + Λ 2 )=

cos 4 4 −

2 .

= (5.115)

Φ

Equiareal (Lambert):

1+cos 4 4 −

2

1+cos 4 2
cos 2 2

Φ

1

2 ( Λ 1 + Λ 2 )= 1

= 1
2

cos 2 4 −

2

. (5.116)

2

Φ

Gnomonic:

cos 2 Δ +1
cos 4 Δ

sin 2 Φ +1
sin 4 Φ

1

2 ( Λ 1 + Λ 2 )= 1

= 1
2

. (5.117)

2

Orthographic:

1

2 ( Λ 1 + Λ 2 )= 1 2 (1 + cos 2 Δ )= 1 2 (1 + sin 2 Φ ) .

(5.118)

Lagrange conformal:

1

2 ( Λ 1 + Λ 2 )= 1

1
cos 4 2

= 1
4

1
cos 4 ( 4 −

2 ) .

(5.119)

4

Φ

Box 5.19 (Total surface element S or the area of a spherical cap).
S = R 2 2 π
0

d Λ π/ 2
Φ

d Φ cos Φ = R 2 2 π
0

d Λ Δ
0

d Δ sin Δ,

(5.120)

subject to

Δ := π

2 − Φ or Φ = π 2 − Δ,

d Φ = − d Δ, (5.121)

π/ 2 0

d Δ =+ Δ
0

d Φ = − d Δ ;

Φ Δ

S =2 πR 2 [+ sin Φ ] π/ Φ =2 πR 2 (1 − sin Φ ) ,

(5.122)

S =2 πR 2 [ cos Δ ] 0 =2 πR 2 (1 − − cos Δ ) .