“Ellipsoid-of-Revolution to Cone”: Polar Aspect - Map Projections: Cartographic Information Systems - page 454

⇔ (19.10)

d f
f

= n

(1 − E 2 cos 2 Δ )sin Δ ⇒ ln f = n (1 − E 2 )

(1 − E 2 )d Δ d Δ

(1 − E 2 cos 2 Δ )sin Δ .

Here, let us substitute u := E cos Δ :

E 2 ) E sin 2 Δ = E 2

u 2

− d u

d Δ = E (

d u

, − sin Δ ) , ln f = − n (1 −

u 2 ) . (19.11) E 2 (1 − u 2 )( E 2 −

By integration-by-parts, we find in detail:

u 2 ) = A d u u 2 + B d u

d u

u 2 ,

(19.12)

(1 − u 2 )( E 2 − 1 − E 2 −

A ( E 2 u 2 )+ B (1 u 2 )=1 , AE 2 + B u 2 ( A + B )=1 , − − −

AE 2 + B =1 , A + B =0
⇒

1 1

A = −

E 2 , B =

1 − 1 − E 2

⇒ (19.13)

1 − E 2

− u 2 =

d u
(1 − u 2 )( E 2

1 d u

1 − u 2 −

d u

− u 2 ) = − E 2

E 2 1

=

1

2 ln 1+ u 2 E ln E + u

1

= −

u −

1 − 1 − E − u

ln 1+ E cos Δ
1

.

E ln 1 − cos Δ

1

E cos Δ + 1

= −

2(1 − E 2 ) − 1+cos Δ

We take advantage of the addition theorem:

ln 1+ E cos Δ +ln c

d Δ
(1 − E 2 cos 2 Δ )sin Δ =

1
2(1 − E 2 ) 1 − E cos Δ + 1 E lntan 2 Δ

−

2

⇒ (19.14)

+ln c =ln 1+ E cos Δ
1 2 c

ln 1+ E cos Δ
1

En/ 2

ln f = nE
2 E cos Δ tan 2 /E Δ

tan n Δ

− 2 − E cos Δ

⇒

f ( Δ ):= c 1+ E cos Δ
1 − E cos Δ

n

E/ 2

tan Δ
2

. (19.15)

Let us here also summarize the general form of the conformal mapping equations as well as the
principal stretches.