Ellipsoid-of-Revolution to Tangential Plane - Map Projections: Cartographic Information Systems - page 269

Box 8.7 (Equiareal mapping of the ellipsoid-of-revolution to the tangential plane at the North
Pole).

Postulate of an areomorphism:
Λ 1 Λ 2 =1 ,

f ( Δ ) √ 1 f ( Δ )(1 E 2 cos 2 Δ ) 3 / 2

A 1 (1

E 2 cos 2 Δ
A 1 sin Δ

− −

=1 ⇒ f d f (8.66)

− E 2 )

1 − E 2

= A 1

E 2 cos 2 Δ ) 2 sin Δ d Δ.

(1 −

Integration of the characteristic differential equations
of a conformal mapping

2 2

A 1 ,A 2

E

A 1 ,A 2 → T N E

:

2 f 2 = A 1

1 − E 2

1

E 2 cos 2 Δ ) 2 sin Δ d Δ + c.

(8.67)

(1 −

Decomposition into rational partials:
y = E cos Δ

sin Δ d( E cos Δ ) d y

1
E

1
E E 2 cos 2 Δ ) 2 d Δ =

−

E 2 cos 2 Δ ) 2 =

−

y 2 ) 2 , (1 − (1 − (1 −

1
(1 − y 2 ) 2 =

A
(1 − y ) 2 +

B
(1 − y ) +

C
(1 + y ) 2 +

D
(1 + y )

A = B = C = D = 1 ⇔

4 .

(8.68)

sin Δ

1
4 E

1 1

E 2 cos 2 Δ ) 2 d Δ = −

y ) 2 +

(1 − (1 − (1 − y )

d y.

1
(1 + y ) 2 +

1
(1 + y )

+

Standard integrals:

ay + b = 1

d y

a ln

| ay + b | ,

d y
(1 + y ) 2 =

1
1+ y ,

d y 1

−

y ) 2 =+ y ,

(8.69)

(1 − 1 −

ln 1+ y =

sin Δ
(1 − E 2 cos 2 Δ ) 2 d Δ =

1
4 E

1
1 − y −

1
1+ y

−

1 − y +

ln 1+ E cos Δ
1

.

1
4 E

1 1

1+ E cos Δ

= −

E cos Δ + E cos Δ −

− 1 −