Ellipsoid-of-Revolution to Tangential Plane - Map Projections: Cartographic Information Systems - page 265

Box 8.6 (Conformal mapping of the ellipsoid-of-revolution to the tangential plane at the
North Pole).

Postulate of conformeomorphism:
Λ 1 = Λ 2 ,

f ( Δ ) √ 1

= f ( Δ )(1

E 2 cos 2 Δ ) 3 / 2
A 1 (1 − E 2 )

E 2 cos 2 Δ
A 1 sin Δ

− − d f

f

⇒

E 2

1 −

=

E 2 cos 2 Δ ) d Δ.

(8.52)

sin Δ (1 −

Integration of the characteristic differential equations
of a conformal mapping:
ln f =

E 2
sin Δ (1 − E 2 cos 2 Δ ) d Δ +ln c.

1 −

(8.53)

Decomposition into rational partials:

E sin Δ
1+ E cos Δ +

,

E 2

1 − 1

sin Δ −

E
2

E sin Δ

E 2 cos 2 Δ ) =

sin Δ (1 − 1 − E cos Δ

E 2

1 −

E 2 cos 2 Δ ) d Δ =lntan Δ

E

2 ln 1

E cos Δ
1+ E cos Δ +ln c =

−

2 − sin Δ (1 −

= artanh(cos Δ ) − E artanh ( E cos Δ )+ln c (8.54)

⇒
f ( Δ )= c 1+ E cos Δ
1

E/ 2

tan Δ

2 ∀

Δ ∈ [0 ,π ]

− E cos Δ

or
f ( Δ )= c exp[artanh(cos Δ )] exp[ − E artanh ( E cos Δ )] .

Integration constant, postulate of isometry at the North Pole:
lim

Δ→ 0 Λ 1 ( Δ )=1 ,

√ 1 − E 2 cos 2 Δ
A 1 sin Δ

Δ→ 0 c 1+ E cos Δ

E/ 2

tan Δ
2

lim =1 , (8.55)

1 − E cos Δ

E/ 2 √ 1

0 Λ 1 ( Δ )= c 1+ E

− E 2 tan( Δ/ 2)

sin Δ

lim
Δ

lim
Δ

=1 .

1 − E A 1

→ → 0

L'Hospital's rule 0 / 0:

tan( Δ/ 2)
sin Δ

(tan( Δ/ 2))
(sin Δ )

lim
Δ→ 0

= lim
Δ→ 0

,