“Sphere to Tangential Plane”: Polar (Normal) Aspect - Map Projections: Cartographic Information Systems - page 220

r
sin 2

R
cos 2 →

= r. (5.101)

Parameterized special perspective mapping (Cartesian coordinates):
x
y

= R tan Δ
2

cos Λ
sin Λ

= R tan π cos Λ

sin Λ

.

Φ
2

4 −

(5.102)

Case(i)( Φ =0):
x
y

= R cos Λ .

(5.103)

sin Λ

Case(ii)( Φ = π/ 2) :
x
y

= 0 .

(5.104)

0

Left principal stretches:

tan 2
2sin 2 cos 2

f ( Δ )
R sin Δ =

1
2cos 2 2

Λ 1 = = ,

(5.105)

Λ 2 = f ( Δ )

1
2cos 2 2

= .

R

Conformality:
Λ 1 = Λ 2 . (5.106)

Left eigenvectors:

1
2cos 2 2

C 1 Λ 1 = E Λ

(Easting) ,

1
2cos 2 2

C 2 Λ 2 = E Φ (Northing) . (5.107)

Left maximal angular distortion:
Σ l = Ψ l − Ψ r =0 ,

(5.108)

Ω l =0 .

Parameterized inverse mapping:

x
x 2 + y 2 ,

cos Λ =

(5.109)

y
x 2 + y 2 ,

sin Λ =