“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 192

R sin Δ = f ( Δ )

f ( Δ ) d f

f

d Δ
sin Δ .

R ⇒

=

Integration of the characteristic differential
equation of a conformal mapping

2
R

2

S → T N S

R :

d y
y sin x =ln

d x

tan x
2

, =ln y,

ln f =ln

d f
f

=

d Δ
sin Δ ⇔

tan Δ
2

+ln c, f ( Δ )= c tan Δ

2 ∀

Δ ∈ ]0 ,π [ . (5.27)

Parameterized conformal mapping:

x
y

=2 R tan Δ
2

cos Λ
sin Λ

=2 R tan π cos Λ

sin Λ

.

Φ
2 4 −

(5.28)

Left principal stretches:

1
cos 2 2

1

cos 2 4 −

2 .

Λ 1 = Λ 2 = = (5.29)

Φ

Left eigenvectors:

1
cos 2 ( 4 −

C 1 Λ 1 = E Λ

2 ) (“Easting”) ,

Φ

(5.30)

1
cos 2 ( 4 −

C 2 Λ 2 = E Φ

2 ) (“Northing”) .

Φ

Left angular shear:
Σ l = Ψ l − Ψ r =0 ,Ω l =0 . (5.31)

Parameterized inverse mapping:
tan Λ = y

2 R x 2 + y 2 . x , tan Δ

1

2 =

(5.32)

Box 5.5 (Distortion analysis at the North Pole).

Postulate of an isometry at the North Pole:
lim

Δ→ 0 Λ 2 ( Δ )=1 .

(5.33)