From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 13

Tab l e 1 . 1 ID card of Hammer retroazimuthal projection of the sphere

(i) Classification Retroazimuthal, modified azimuthal, neither conformal nor equal area.

(ii) Graticule Meridians: central meridian is straight, other meridians are curved.

Parallels: curved. Poles of the sphere: curved lines.
Symmetry: about the central meridians.

(iii) Distortions Distortions of area and shape.

(iv) Other features The direction from any point to the center of the map is the angle that a

straight line connecting the two points makes with a vertical line. This
feature is the basis of the term “retroazimuthal”. Scimitar-shaped
boundary. Considerable overlapping when entire sphere is shown.

(v) Usage To determine the direction of a central point from a given location.

(vi) Origins Presented by E. Hammer (1858-1925) in 1910. The author is the successor

of E. Hammer in the Geodesy Chair of Stuttgart University (Germany). The
map projection was independently presented by E.A. Reeves (1862-1945)
and A.R. Hinks (1874-1945) of England in 1929.

Fig. 1.5. Special map projection of the sphere, called Hammer retroazimuthal projection , centered near St.Louis
(longitude 90 ◦ W, latitude 40 ◦ N), with shorelines, 15 ◦ graticule, two hemispheres, one of which appears backwards
(they should be superimposed for the full map)

left stretch: right stretch:

d s 2

2 d s 2 =d S 2 ,Λ r =: λ 2 = d S 2 Λ 2 d S 2 =d s 2 ,

d S 2 = Λ 2 =: Λ l , d s 2 ,

(1.47)

subject to duality Λ 2 λ 2 =1 .

Λ 2 ,λ 2

Question: “What is the role of stretch in the con-

text of the pair of (symmetric, positive-definite) matrices.
{

{ }

, respec-
tively?” Answer: “Due to a standard lemma of matrix
algebra, both matrices can be simultaneously diagonalized,
one matrix being the unit matrix.”

c MN ,G MN } , { C l , G l } ,and { C μν ,g μν } , { C r , G r }