Generalized Hammer Projection - Map Projections: Cartographic Information Systems - page 802

E 2 sin 2 A ∗ 1

p 2 :=

, (H.98)

1 − E 2 1 − E 2 cos 2 A ∗

A 1 (1 E 2 )

−

r 2 =

E 2 cos 2 A ∗ × 1 −

⎡

√ 1 − E 2 √ 1 − E 2 cos 2 A ∗
E sin A ∗

E 2 sin 2 A ∗ sin 2 Δ ∗

⎣ − cos Δ ∗

× 1+

E 2 cos 2 A ∗ ) −

×

(1 − E 2 )(1 −

1+

E 2 sin 2 A ∗

× × (H.99)

(1 − E 2 )(1 − E 2 cos 2 A ∗ )

⎛ ⎞ ⎤

Δ ∗

E sin A ∗ cos Δ ∗

⎝ ⎠ ⎦

√ 1

E 2 √ 1

1+

× arcsin ,

− − E 2 cos 2 A ∗

E 2 sin 2 A ∗
(1 −E 2 )(1 −E 2 cos 2 A ∗ )

0

A 1 √ 1 − E 2

r 2 =

E 2 cos 2 A ∗ ) 3 / 2 ×

(1 −

cos Δ ∗ (1 E 2 cos 2 A ∗ + E 2 sin 2 Δ ∗ sin 2 A ∗ ) × − − E 2 )(1 − −

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

E cos Δ ∗ sin A ∗

1 −

− arcsin √ 1 + (H.100)

− 2 E 2 cos 2 A ∗ + E 4 cos 2 A ∗

+ (1 − E 2 )(1 − E 2 cos 2 A ∗ )+

,

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

E sin A ∗

+ 1

−

√ 1

arcsin

− 2 E 2 cos 2 A ∗ + E 4 cos 2 A ∗

r = A 1 √ 2 √ 1 − cos Δ ∗ = A 1 √ 2 √ 1 − sin B ∗ ,

(H.101)

arcsin x
x

a

√ 1

lim
x→ 0

lim
x→ 0

a 2 x 2 = a,

(H.102)

−

arcsin( E sin A ∗ cos Δ ∗ )
E sin A ∗

=cos Δ ∗ ,

lim
E→ 0

arcsin( E sin A ∗ )
E sin A ∗

=1 .

lim
E→ 0

(H.103)

A ∗ ,B ∗ )

H-4 The Transformation of the Radial Function r (

Λ ∗ ,Φ ∗ ) into r (

In this section, the transformation of the radial function r ( A ∗ ,B ∗ )into r ( Λ ∗ ,Φ ∗ )ispresented.
The following relations ( H.104 )-( H.112 ) specify this transformation.

First factor (see ( H.14 ), ( H.30 )) :

A 1 √ 1 − E 2 (cos 2 ( Λ ∗ − Ω )cos 2 Φ ∗ +(1 − E 2 ) 2 sin 2 Φ ∗ ) 3 / 2

(cos 2 ( Λ ∗ −

A 1

E 2 )sin 2 Φ ∗ ) 3 / 2 . E 2 cos 2 A ∗ ) 3 / 2 =

(H.104)

(1 − (1 − E 2 ) Ω )cos 2 Φ ∗ +(1 −