“Ellipsoid-of-Revolution to Cone”: Polar Aspect - Map Projections: Cartographic Information Systems - page 461

A 1 sin Δ i Λ

(1 − E 2 cos 2 Δ i ) 1 / 2 = f ( Δ i ) nΛ

∀ i ∈{ 1 , 2 } , (19.44)

f ( Δ i )= c 2

1 / 2

A 1 (1 E 2 )

− cos Δ i

2 E ln 1+ E cos Δ i

1

−

E 2 cos 2 Δ i + n 1 − 1 − E cos Δ i

⇒

A 1 sin 2 Δ i
n 2 (1

A 1 (1 E 2 )

− cos Δ i

2 E ln 1+ E cos Δ i

1

E 2 cos 2 Δ i ) = c 2

−

E 2 cos 2 Δ i +

(19.45)

− n 1 − 1 − E cos Δ i

⇒

sin 2 Δ i
n (1 − E 2 cos 2 Δ i ) +(1

E 2 )

1 / 2

c = A 1

cos Δ i
1 − E 2 cos 2 Δ i + 2 E ln 1+ E cos Δ i

1

√ n − ,

1 − E cos Δ i

c ( Δ 1 )= c ( Δ 2 ) .
Let us substitute the two functions h ( Δ i )and g ( Δ i ):

sin 2 Δ i

h ( Δ i )= h i :=

E 2 cos 2 Δ i ,

(19.46)

1 −

g ( Δ i )= g i := (1 − E 2 )

.

cos Δ i

2 E ln 1+ E cos Δ i

1

E 2 cos 2 Δ i +

1 − 1 − E cos Δ i

For c , we then arrive at

h i

n + g i 1 / 2

c = A 1 √ n .

(19.47)

For n, we then arrive at

h 1

n + g 1 1 / 2

h 2

n + g 2 1 / 2

A 1
√ n = A 1 √ n

⇒
n = h 1

− h 2

(19.48)

g 2 − g 1

⇒

sin 2 Δ 1 sin 2 Δ 2

1 −E 2 cos 2 Δ 2 1 −E 2 cos 2 Δ 1 −

E 2 ) + 2 E ln 1+ E cos Δ 2

.

n = (19.49)

cos Δ 2 cos Δ 1

1 −E 2 cos 2 Δ 2

1 − E cos Δ 1
1+ E cos Δ 1

(1 −

1 −E 2 cos 2 Δ 2 − 1 −E cos Δ 2