“Ellipsoid-of-Revolution to Cylinder”: Polar Aspect - Map Projections: Cartographic Information Systems - page 340

1

E 2 sin 2 Φ
cos Φ

E 2 sin 2 Φ ) 3 / 2
A 1 (1

−

= (1

−

f ( Φ ) Λ 1 = Λ 2 ⇔ ⇒

− E 2 )

(14.16)

E 2 sin 2 Φ d Φ ⇒ f ( Φ )= A 1 Φ

⇒ d f = A 1 (1

E 2 )
cos Φ

− 1 d Φ

cos Φ

1 − E 2

E 2 sin 2 Φ . 1 − 1 −

0

The integral is called “isometric latitude”. Applying “integration-by-parts”, we obtain the follow-
ing set of formulae.
f ( Φ )= A 1 Φ
0

d Φ 1

E cos Φ
1+ E sin Φ

,

E cos Φ
1 − E sin Φ

E
2 cos Φ −

+ (14.17)

f ( Φ )= A 1 ln tan π

4 + Φ

A 1 E
2

ln 1+ E sin Φ
1

−

E sin Φ .

(14.18)

2 −

At this point, let us present the mapping equations as well as the principal stretches. They are
easily computed as follows.

=

,

x
y

A 1 Λ

A 1 ln tan 4 + 2 1 − E sin Φ

1+ E sin Φ E/ 2

(14.19)

Λ 1 = Λ 2 = 1

E 2 sin 2 Φ
cos Φ

−

. (14.20)

14-23 Special Normal Cylindric Mapping (Normal Equiareal,
Equidistant: Equator)

Second, let us here apply the postulate of equiareal mapping. In doing so, we obtain the following
chain of relations.

1

E 2 sin 2 Φ
cos Φ

f ( Φ )
A 1 (1

−

E 2 sin 2 Φ ) 3 / 2 =1 Λ 1 Λ 2 =1 ⇔

E 2 ) (1

− ⇒

−

(14.21)

E 2 sin 2 Φ ) 2 d Φ ⇒ f ( Φ )= A 1 (1 − E 2 ) Φ

cos Φ cos Φ

⇒ d f = A 1 (1 − E 2 )

d Φ

E 2 sin 2 Φ ) 2 , (1 − (1 −

0