“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 233

Fig. 5.30. Wiechel's pseudo-azimuthal projection “sphere to plane”, normal aspect

Box 5.22 (Wiechel's pseudo-azimuthal projection “sphere to plane”).
Direct equations of Wiechel's map (polar coordinates):

2 = Λ + π

=: g ( Λ, Δ ) ,

α = Λ + Δ

Φ
2 4 −

(5.147)

2 =2 R sin π

=: f ( Δ ) .

r =2 R sin Δ

Φ
2

4 −

Direct equations of Wiechel's map (Cartesian coordinates):
x
y

=2 R sin Δ
2

cos( Λ + 2 )
sin( Λ + 2 )

, x = f ( Δ ) cos g ( Λ, Δ ) .

(5.148)

y sin g ( Λ, Δ )

Left Jacobi matrix:

J l = D Λ xD Δ x

− ,f = R cos Δ

f sin gf cos g

2 ,g Δ = 1

fg Δ sin g
+ f cos gf sin g + fg Δ cos g

−

2 .

(5.149)

D Λ yD Δ y

Left Cauchy-Green matrix:

G r =I 2 , C l =J l G r J l =J l J l = f 2

1

2 f 2

1

2 f 2 f 2 + f 2 g Δ

= 4 R 2 sin 2 2 2 R 2 sin 2 2

.

(5.150)

2 R 2 sin 2 2 R 2