Ellipsoid-of-Revolution to Sphere and from Sphere to Plane - Map Projections: Cartographic Information Systems - page 295

d KL = −

,

a 2 r cos 2 φ

0

(9.35)

rf 2 ( Φ ) 0 −

a 2 r cos 2 φ
A 1 cos 2 Φ (1

d Λ 1 = d 11 /H 11 = E 2 sin 2 Φ ) 1 / 2 ,

−

d Λ 2 = d 22 /H 22 =

rf 2 ( Φ )
A 1 (1

E 2 sin 2 Φ ) 3 / 2 .

E 2 ) (1

− (9.36)

−

9-2 The Conformal Mappings “Ellipsoid-of-Revolution to Plane”

The conformal mappings from the ellipsoid-of-revolution to the plane: the conditions of con-
formality, the standard integrals, spherical isometric latitude, ellipsoidal isometric latitude.

First, we postulate the condition of conformality Λ 1 = Λ 2 and subsequently we take advantage of
the standard integrals that are collected in Box 9.6 .

Λ 1 = Λ 2
⇔

ar cos φ
A 1 cos Φ (1

r
A 1 (1 − E 2 )

d φ
d Φ (1

E 2 sin 2 Φ ) 1 / 2 = E 2 sin 2 Φ ) 3 / 2

− − (9.37)

⇔

E 2
1 − E 2 sin 2 Φ

d φ
cos φ =

1 − a

cos Φ d Φ.

(9.38)

We here note in passing that via the first standard integral, we introduce spherical isometric
latitude . By integration-by-parts, we split the second standard integral into three parts, namely
by introducing ellipsoidal isometric latitude .

Box 9.6 (The standard integrals).

First standard integral:

d φ

cos φ = I first =lntan π

:= q

4 + φ

(9.39)

2