Generalized Hammer Projection - Map Projections: Cartographic Information Systems - page 801

(iv)

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

A 1

S subject to c 1 =2 ,c 2 =1 ,c 3 =1 / 2, and c 4 =1,namely

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

(H.93)

cos Φ sin Λ/ 2 sin Φ

1 1

cos α ( Λ, Φ )=

cos 2 Φ cos 2 Λ/ 2 , sin α ( Λ, Φ )= cos 2 Φ cos 2 Λ/ 2 ,

(H.94)

− −

r = A √ 2 1 − cos Φ cos Λ/ 2 , (H.95)

x =2 A 1 √ 2 1+cos Φ cos Λ/ 2 , y = A 1 √ 2

cos Φ sin Λ/ 2 sin Φ

1+cos Φ cos Λ/ 2 .

(H.96)

End of Corollary.

As a visualization for the derived mapping equations for the ellipsoidal Hammer projection ,at
the beginning of this section, Fig. H.3 is given including the Tissot indicatrices .

Fig. H.3. The ellipsoidal Hammer projection, squared relative eccentricity E 2 =0 . 1

H-3 An Integration Formula

Equation ( H.28 ) may be written as an integration formula. The following relations ( H.97 )-( H.100 )
specify this integration formula. If the relative eccentricity approaches E = 0, then the radial
coordinate specializes to ( H.101 ) according to the L'Hopital Rule ( H.102 )and( H.103 ).
sin x 1+ p 2 sin 2 x d x = −

2 cos x 1+ p 2 sin 2 x −

1 1+ p 2

2 p

p cos x
1+ p 2 ,

arcsin (H.97)

subject to