From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 90

cos φ d φ =sin φ + c l .

Boundary conditions:
Φ =0 ⇔ φ =0 ⇒ c l = c r =0 . (1.280)

2 2

Equiareal map of E

A 1 ,A 1 ,A 2 → S r :

;
case 1 : A 1 = r ;case2: left global surface element coincides
with right global surface element

r 2 sin φ = A 1 (1 − E 2 )

sin Φ

4 E ln 1+ E sin Φ

1

E 2 sin 2 Φ ) + 2(1 − 1 − E sin Φ

;

2
A 1 ,A 1 ,A 2

2

S l =area( E )=area( S

r )= S r ;

(1.281)

4 πA 1 1

=4 πr 2

E 2
2 E 2 + 1

−

ln 1+ E
1 − E

⇒

2 A 1 1+ 1

.

E 2
2 E

r 2 = 1

−

ln 1+ E
1 − E

Authalic latitude ( Adams 1921 , p. 65; Snyder 1982 , p. 19):

φ = f ( Φ ) ,
sin φ =sin f ( Φ | S l = S r ) . (1.282)

2 2

Box 1.42 (The authalic equiareal map: E

A 1 ,A 1 ,A 2 → S r ).

Authalic equiareal map:
λ = Λ,

E 2 )

/ 1+ 1 .

E 2
2 E

sin Φ 1

2 E

1+ E sin Φ
1

−

ln 1+ E
1

sin φ =(1 −

E 2 sin 2 Φ +

(1.283)

− E sin Φ − E

1 −

Left and right principal stretches:

A 1 cos Φ 1

r cos φ

E 2 sin 2 Φ,

Λ 1 = −

r

E 2 ) f ( Φ )(1

E 2 sin 2 Φ ) 3 / 2 ,

Λ 2 = −

A 1 (1 −

E 2 )
(1 − E 2 sin 2 Φ ) 2 cos Φ,

A 1 (1

1
r 2 cos φ

−

φ = f ( Φ )= φ = f ( Φ ) ⇒ (1.284)

A 1 cos φ 1

r cos φ

Λ 1 = λ − 1 = E 2 sin 2 Φ,

−