Generalized Hammer Projection - Map Projections: Cartographic Information Systems - page 793

The mapping of the biaxial ellipsoid onto the transverse tangent plane normal to E 3 is equiareal if

(i)
α = arctan (1

E 2 )tan Φ ∗ −

Ω ) , − cos( Λ ∗ −

r = A 1 cos 2 ( Λ ∗ −

(cos 2 ( Λ ∗ − Ω )cos 2 Φ ∗ +(1 − E 2 )sin 2 Φ ∗ ) 3 / 4 t 1 + t 2 + t 3 + t 4

E 2 ) 2 sin 2 Φ ∗

Ω )cos 2 Φ ∗ +(1 −

(H.33)

(in polar coordinates) ,
subject to

cos 2 ( Λ ∗ −

sin( Λ ∗ − Ω )cos Φ ∗

t 1 = E 2 )sin 2 Φ ∗ , − Ω )cos 2 Φ ∗ +(1 −

E 2 sin 2 Φ ∗

1 −

sin 2 ( Λ ∗ − Ω )cos 2 Φ ∗

E sin Φ ∗

E sin( Λ ∗ − Ω )sin Φ ∗ cos Φ ∗

1 −

t 2 =

− arcsin (1

Ω )cos 2 Φ ∗ ) ,

(H.34)

E 2 sin 2 Φ ∗ )(1 sin 2 ( Λ ∗ −

− −

t 3 = cos 2 ( Λ ∗ − E 2 )sin 2 Φ ∗ ,

Ω )cos 2 Φ ∗ +(1 −

t 4 = 1 − sin 2 ( Λ ∗ − Ω )cos 2 Φ ∗

E sin Φ ∗

arcsin 1 ,

E sin Φ ∗

sin 2 ( Λ ∗ −

− Ω )cos 2 Φ ∗

(ii)

cos( Λ ∗ −

− Ω )

cos 2 ( Λ ∗ −

cos α = , (H.35)

E 2 ) 2 tan 2 Φ ∗ Ω )+(1 −

E 2 )tan Φ ∗ (1 −

cos 2 ( Λ ∗ −

sin α =

E 2 ) 2 tan 2 Φ ∗
(in Cartesian coordinates x ∗ = r cos α and y ∗ = r sin α ) ,

Ω )+(1 −

(iii)

tan Φ ∗

α = lim

E→ 0 α ( E ) = arctan

Ω ) ,

(H.36)

− cos( Λ ∗ −

E→ 0 r ( E )= A 1 √ 2 1 − sin( Λ ∗ − Ω )cos Φ ∗
(if E =0) .

r = lim

End of Lemma.

Corollary H.4 (Equiareal mapping of the biaxial ellipsoid onto the transverse tangent plane,
special case Ω =3 π/ 2).

The mapping of the biaxial ellipsoid onto the transverse tangent plane normal to E 3 is equiareal if

(i)
α = arctan (1

− E 2 )tan Φ ∗

sin Λ ∗

,