Ellipsoid-of-Revolution to Tangential Plane - Map Projections: Cartographic Information Systems - page 255

Box 8.3 (Equidistant mapping of the ellipsoid-of-revolution to the tangential plane at the
North Pole).

Parameterized mapping:
α = Λ, r = f ( Δ ) ,Δ := π/ 2 − Φ, (8.20)

x = r cos α = f ( Δ )cos Λ, y = r sin α = f ( Δ )sin Λ.

Canonical postulate Λ 2 =1 ,
equidistant mapping of the family of meridians:
Λ 2 = f ( Δ ) (1 − E 2 cos 2 Δ ) 3 / 2 =1

A 1 (1 − E 2 )

⇔

d Δ

E 2 )

d f = A 1 (1 − (8.21)

(1 − E 2 cos 2 Δ ) 3 / 2

⇒
f ( Δ )= A 1 (1 − E 2 ) Δ
0

d Δ

E 2 cos 2 Δ ) 3 / 2 .

(1 −

Transformation of surface normal latitude Φ to reduced latitude Φ ∗ :
tan Φ ∗ = √ 1

− E 2 tan Φ

⇔ (8.22)

1

√ 1 − E 2 tan Φ ∗ . tan Φ =

Equidistant mapping of the family of meridians,
elliptic integral of the second kind:
f ( Δ ) → f ( Φ ) ,

E 2 ) π/ 2
π/ 2 −Φ

d Φ
(1 − E 2 sin 2 Φ ) 3 / 2 ;
f ( Φ ) → f ( Φ ∗ ) ,
f ( Φ ∗ )= A 1 π/ 2
π/ 2 −Φ

f ( Φ )= A 1 (1 −

1

E 2 cos 2 Φ ∗ d ∗ ; − (8.23)

f ( Φ ∗ ) f ( Δ ∗ ) ,

→

f ( Δ ∗ )= A 1 Δ ∗
0

1

E 2 sin 2 Δ d Δ =: A 1 E ( Δ ∗ ,E ) .

−

Elliptic integral of the second kind:
f ( Φ )= A 1 E π/ 2 − arc tan √ 1 − E 2 tan Φ ,E . (8.24)