From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 41

Fig. 1.18. Special map projection of the sphere: the Hammer equiareal modified azimuthal projection. This map
projection is centered to the Greenwich meridian, with shorelines, 30 ◦ longitude, 15 ◦ latitude graticule, Tissot
ellipses of distortion, “the world in one chart”

A map projection which is worth studying with all the machinery of deformation measures is
the Hammer equiareal modified azimuthal projection of the sphere

2

S

R + presented in Fig. 1.18 .The

ID card of this special map projection is shown in Table 1.2 .

Tab l e 1 . 2 ID card of Hammer equiareal modified azimuthal projection of the sphere

(i) Classification Modified azimuthal, transverse, rescaled equiareal. 1.0
(ii) Graticule Meridians: central meridian is straight, other meridians are
algebraic curves of fourth order. The limiting meridians
form an ellipse.
Parallels: curved. equator is straight, other parallels
are algebraic curves of fourth order.
Poles of the sphere: points.
Symmetry: about the central meridians.
(iii) Distortions Product of principal stretches is one, equiareal,
equidista n t map of th e equator.
(iv) Direct mapping equations x = c 1 R √ 2 √ 1 − c 4 sin 2 Φ sin( c 3 Λ ) ,

1+ √ 1 −c 4 sin 2 Φ cos( c 3 Λ )

c 2 R √ 2 c 4 sin Φ

y = ,

1+ √ 1 −c 4 sin 2 Φ cos( c 3 Λ )

c 1 =2, c 2 =1,
c 3 = 2 , c 4 =1,

c 1 c 2 c 3 c 4 =1 .

(v) Usage Atlas cartography.

(vi) Origins Presented by E. Hammer (1858-1925) in 1892. The special

Hammer projection has been generalized from the sphere
S 2 R + to the ellipsoid-of-revolution E 2 A 1 ,A 1 ,A 2

by Grafarend and Syffus ( 1997e ).