Generalized Hammer Projection - Map Projections: Cartographic Information Systems - page 794

r = A 1 sin 2 Λ ∗ 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 ∗

−

(H.37)

(sin 2 Λ ∗ cos 2 Φ ∗ +(1

−

(in polar coordinates) ,
subject to

sin 2 Λ ∗ cos 2 Φ ∗ +(1 − E 2 )sin 2 Φ ∗ ,

cos Λ ∗ cos Φ ∗
1

t 1 = −

E 2 sin 2 Φ ∗

−

1 − cos 2 Λ ∗ cos 2 Φ ∗
E sin Φ ∗

E cos Λ ∗ sin Φ ∗ cos Φ ∗

t 2 = −

(1

cos 2 Λ ∗ cos 2 Φ ∗ ) ,

arcsin (H.38)

E 2 sin 2 Φ ∗ )(1

− −

t 3 = sin 2 Λ ∗ cos 2 Φ ∗ +(1

− E 2 )sin Φ ∗ ,

t 4 = 1 − cos 2 Λ ∗ cos 2 Φ ∗

E sin Φ ∗

√ 1

arcsin ,

E sin Φ ∗

− cos 2 Λ ∗ cos 2 Φ ∗

(ii)

sin Λ ∗
sin 2 Λ ∗ +(1 − E 2 ) 2 tan 2 Φ ∗

E 2 )tan Φ ∗
sin 2 Λ ∗ +(1 − E 2 ) 2 tan 2 Φ ∗

(1 −

cos α = , sin α = (H.39)

(in Cartesian coordinates x ∗ = r cos α and y ∗ = r sin α ) ,

(iii)

E→ 0 r ( E )= A 1 √ 2 √ 1

E→ 0 α ( E ) = arctan tan Φ ∗

α = lim

sin Λ ∗ , r = lim

− cos Λ ∗ cos Φ ∗ (H.40)

(if E =0) .

End of Corollary.

H-2 The Ellipsoidal Hammer Projection

The second constituent of the Hammer projection is a proper change of scale of the transverse
equiareal projection which conserves the local area.

Section H-21.

As a starting point, we set up in Sect. H-21 the equations of an equiareal mapping from a left
biaxial ellipsoid to a right biaxial ellipsoid in order to be motivated for the structure of a change
of scale.

Section H-22.

Section H-22 introduces in detail the rescaled equations x = c 1 x ∗ ( Λ ∗ ,Φ ∗ ) y = c 2 y ∗ ( Λ ∗ ,Φ ∗ )ofa
transverse equiareal projection with respect to a right biaxial ellipsoid

2
A 1 ,A 2

E . Surface normal