Generalized Hammer Projection - Map Projections: Cartographic Information Systems - page 800

(ii)

If the relative eccentricity vanishes, E = 0, then we arrive at the Hammer projection of the sphere
S

A 1

,namely

x = c 1 r ( Λ, Φ ; c 3 ,c 4 )cos α ( Λ, Φ ; c 3 ,c 4 ) ,
y = c 2 r ( Λ, Φ ; c 3 ,c 4 )sin α ( Λ, Φ ; c 3 ,c 4 ) , (H.83)

subject to

1

c 4 sin 2 Φ sin c 3 Λ

−

cos α ( Λ, Φ ; c 3 ,c 4 )= 1 , (H.84)

c 4 sin 2 Φ )cos 2 c 3 Λ

− (1 −

c 4 sin Φ
1 − (1 − c 4 sin 2 Φ )cos 2 c 3 Λ sin α ( Λ, Φ ; c 3 ,c 4 )= , (H.85)

r = A 1 √ 2 1 −

1 − c 4 sin 2 Φ cos c 3 Λ,

(H.86)

1 − c 4 sin 2 Φ sin c 3 Λ

x = c 1 A 1 √ 2 1+ 1 , (H.87)

c 4 sin 2 Φ cos c 3 Λ

−

y = c 2 A 1 √ 2

c 4 sin Φ

1+ 1 . (H.88)

c 4 sin 2 Φ cos c 3 Λ −

(iii)

If we choose c 1 =2 ,c 2 =1 ,c 3 =1 / 2, and c 4 = 1 which fulfills c 1 c 2 c 3 c 4 = 1 (Hammer's choice),
then the mapping of the right biaxial ellipsoid

2
A 1 ,A 2

E with respect to left biaxial ellipsoid E

A 1 ∗ ,A 2 ∗
subject to A 1 ∗ = A 1 ,A 2 ∗ = A 2 onto the transverse tangent plane being normal to E 3 and rescaled,
namely of equiareal type, reduces to

x =2 r ( Λ, Φ )cos α ( Λ, Φ ) , y = r ( Λ, Φ )sin α ( Λ, Φ ) ,

(H.89)

sin Λ/ 2

sin 2 Λ/ 2+(1

cos α ( Λ, Φ )=

E 2 ) 2 tan 2 Φ ,

(H.90)

−

E 2 )tan Φ

(1 −

sin 2 Λ/ 2+(1

sin α ( Λ, Φ )=

E 2 ) 2 tan 2 Φ , −

sin 2 Λ/ 2cos 2 Φ +(1 E 2 ) 2 sin 2 Φ −

r 2 ( Λ, Φ )= A 1

E 2 )sin 2 Φ ) 3 / 2 ( t 1 + t 2 + t 3 + t 4 ) ,

(H.91)

(sin 2 Λ/ 2cos 2 Φ +(1

−

sin 2 Λ/ 2cos 2 Φ +(1

cos Λ/ 2cos Φ
1 − E 2 sin 2 Φ

t 1 = E 2 )sin 2 Φ,

− −

cos 2 Λ/ 2cos 2 Φ
E 2 sin Φ

1 − E sin Φ cos Φ cos Λ/ 2

(1 − E 2 sin 2 Φ )(1 − cos 2 Λ/ 2cos 2 Φ ) t 2 =

− arcsin , (H.92)

t 3 = sin 2 Λ/ 2cos 2 Φ +(1 − E 2 )sin 2 Φ,

cos 2 Λ/ 2cos 2 Φ
E sin Φ

t 4 = 1

− E sin Φ

arcsin 1

cos 2 Λ/ 2cos 2 Φ . −