Generalized Hammer Projection - Map Projections: Cartographic Information Systems - page 803

First term (see ( H.12 ), ( H.14 ), ( H.15 ), ( H.30 )) :

sin B ∗ (1 t 1 = t 1 ( A ∗ ,B ∗ ):=

E 2 cos 2 A ∗ )+ E 2 cos 2 B ∗ sin 2 A ∗ ,

− − E 2 )(1 − (H.105)

sin( Λ ∗ − Ω )cos Φ ∗ (1 − E 2 )
1

t 1 = t 1 ( Λ ∗ ,Φ ∗ ):= −

×

E 2 sin 2 Φ ∗ −

cos 2 ( Λ ∗ −

E 2 )sin 2 Φ ∗

Ω )cos 2 Φ ∗ +(1 −

cos 2 ( Λ ∗ −

× . (H.106)

E 2 ) 2 sin 2 Φ ∗

Ω )cos 2 Φ ∗ +(1 −

Second term (see ( H.12 ), ( H.14 ), ( H.15 ), ( H.30 )) :

2 E 2 cos 2 A ∗ + E 4 cos 2 A ∗
E sin A ∗

1 −

t 2 = f 2 ( A ∗ ,B ∗ ):=

− ×

E sin B ∗ sin A ∗
√ 1 − 2 E 2 cos 2 A ∗ + E 4 cos 2 A ∗ × arcsin , (H.107)

E 2
E sin Φ ∗

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

sin 2 ( Λ ∗ −

1 − 1 −

t 2 = t 2 ( Λ ∗ ,Φ ∗ ):= − (H.108)

E sin( Λ ∗ − Ω )sin Φ ∗ cos Φ ∗

(1

× arcsin

Ω )cos 2 Φ ∗ ) .

E 2 sin 2 Φ ∗ )(1 sin 2 ( Λ ∗ −

− −

Third term (see ( H.14 ), ( H.30 )) :

t 3 = t 3 ( A ∗ ,B ∗ ):= √ 1 − E 2 √ 1 − E 2 cos 2 A ∗ , (H.109)

t 3 = t 3 ( Λ ∗ ,Φ ∗ ):=(1 − E 2 ) cos 2 ( Λ ∗ ,Ω ∗ )cos 2 Φ ∗ +(1

E 2 )sin 2 Φ ∗

−

cos 2 ( Λ ∗ ,Ω ∗ )cos 2 Φ ∗ +(1 . (H.110)

E 2 ) 2 sin 2 Φ ∗

−

Fourth term (see( H.14 ), ( H.30 )) :

2 E 2 cos 2 A ∗ + E 4 cos 2 A ∗
E sin A ∗

E sin A ∗

t 4 = t 4 ( A ∗ ,B ∗ ):= 1

−

arcsin √ 1 , (H.111)

− 2 E 2 cos 2 A ∗ + E 4 cos 2 A ∗

E 2
E sin Φ ∗ ×

t 4 = t 4 ( Λ ∗ ,Φ ∗ ):= 1

−

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

sin 2 ( Λ ∗ − E sin Φ ∗

1 − sin 2 ( Λ ∗ − Ω )cos 2 Φ ∗

1 −

× arcsin . (H.112)

H-5 The Inverse of a Special Univariate Homogeneous Polynomial

In order to solve ( H.65 )forsin Φ ∗ , we proceed to present the series expansions of ar tanh x
in ( H.113 )andof(1+ x ) − 1 in ( H.114 ) (compare with Abramowitz and Stegun 1965 ). Those series
expansions is applied to the two terms ( H.115 )and( H.116 ) which appear in ( H.65 ). In particular,