Ellipsoid-of-Revolution to Sphere and from Sphere to Plane - Map Projections: Cartographic Information Systems - page 302

Box 9.7 (The standard integrals).

First standard integral

(conic mapping: Lambert conformal):

cos Φ d Φ

r 2 sin φ = A 1 (1

E 2 )

−

(1 − E 2 sin 2 Φ ) 2 + c, a

Δ := π

2 −

Φ ⇒− d Δ =+d Φ, c := 0 ,

sin Δ cos Δ

4 E ln 1+ E cos Δ

1

−

E 2 cos 2 Δ ) 2 d Δ = E 2 cos 2 Δ ) + E cos Δ =

(9.70)

(1 − 2(1 − 1 −

sin Φ

4 E ln 1+ E sin Φ

1

E sin Φ .

=

E 2 sin 2 Φ ) +

2(1 − 1 −

Second standard integral

(equal area conic mapping):

E 2 )

,

sin Φ

4 E ln 1+ E sin Φ

1

ar 2 sin φ = A 1 (1

−

E 2 sin 2 Φ ) +

1 − E sin Φ

2(1 −

.

sin φ = A 1 (1

E 2 )

− sin Φ

4 E ln 1+ E sin Φ

1

E 2 sin 2 Φ ) +

(9.71)

ar 2

1 − E sin Φ

2(1 −

The term authalic latitude φ has been introduced by Adams ( 1921 , p. 65) or Snyder ( 1982 , p. 19).
In Box 9.8 , the two interesting concepts are summarized.

Box 9.8 (The two interesting concepts).

First case:
A 1 = r, a =1 . (9.72)

Second case
(“identical surface area”):
4 πr 2 =4 πA 1 1

2 + 1 − E 2

ln 1+ E
1 4 E − E

⇒

2 A 1 1+ 1

E 2
2 E

r 2 = 1

−

ln 1+ E
1

(9.73)

− E

⇒