“Ellipsoid-of-Revolution to Cone”: Polar Aspect - Map Projections: Cartographic Information Systems - page 458

f ( Δ )
M

nf ( Δ )
N sin Δ

Λ 1 Λ 2 =1 ⇒ =1

⇒

E 2 )

f d f = MN sin Δ
n

A 1 (1 − A 1

, M :=

E 2 cos 2 Δ ) 3 / 2 , N :=

(19.32)

(1 − (1 − E 2 cos 2 Δ ) 1 / 2

⇒

f d f =

E 2 )sin Δ
(1 − E 2 cos 2 Δ ) 2 n d Δ

A 1 (1 − 1

2 f 2 = A 1 (1

− E 2 ) sin Δ

(1 − E 2 cos 2 Δ ) 2 d Δ.

⇒

n

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

E 2 cos 2 Δ ) 2 d Δ =

sin Δ

1

−

u 2 ) 2 d u =

(1 − E (1 −

u
2 E (1 − u 2 ) + 4 E ln 1+ u

1 cos Δ

2(1 − E 2 cos 2 Δ ) + 4 E ln 1+ E cos Δ

1

1 − u + c = 1 − E cos Δ + c

=

⇒

A 1 (1 E 2 ) 1

2 f 2 = 1

− cos Δ

2 E ln 1+ E cos Δ

1

2 c 2

−

E 2 cos 2 Δ +

(19.33)

2 n 1 − 1 − E cos Δ

⇒

f = c 2

1 / 2

A 1 (1 − E 2 ) cos Δ

2 E ln 1+ E cos Δ

1

−

E 2 cos 2 Δ +

.

n 1 − 1 − E cos Δ

We fix the integration constant:

Δ = π

2 :
r = f ( π/ 2) = c. (19.34)

We summarize the general form of the mapping equations:

=

1 −E cos Δ 1 / 2 .

α
r

nΛ

c 2

(19.35)

A 1 E 2 ) (1 −

1 −E 2 cos 2 Δ + 2 E ln 1+ E cos Δ

cos Δ

−

n

19-231 Equiareal Mapping: The Variant of Type Equidistant and Conformal on the Reference
Circle

The projection constant n is fixed by the postulate of an equidistant and conformal mapping on
the reference circle Φ 0 .