614 H Generalized Hammer projection
Lemma H.5 (The equiareal mapping of the biaxial ellipsoid with respect to a transverse frame of reference
and a change of scale (the Hammer projection of E
2
A 1 ,A 2
)).
The mapping of the right biax ial ellipsoid E
2
A 1 ,A 2
with respect to left biax ial ellipsoid E
2
A 1∗ ,A 2∗
subject
to A 1∗ = A 1 , A 2∗ = A 2 onto the transverse tangent plane normal to E 3 and with respect to a change
of scale is eq uiareal if
x = c 1 r(Λ, Φ; c 3 , c 4 ) cos α(Λ, Φ; c 3 , c 4 ) ,
(H.6 8)
y = c 2 r(Λ, Φ; c 3 , c 4 ) sin α(Λ, Φ; c 3 , c 4 ) ,
(H.6 9)
subject to
cos α(Λ, Φ; c 3 , c 4 ) =
=
− cos[Λ
∗ (Λ; c 3 ) − Ω]
cos 2 [Λ ∗ (Λ; c 3 ) − Ω]+ (1 − E 2 ) 2 tan
2 [Φ ∗ (Φ; c 4 )]
,
(H.70)
sin α(Λ, Φ; c 3 , c 4 ) =
=
(1 − E
2 ) tan[Φ
∗ (Φ; c 4 )]
cos 2 [Λ ∗ (Λ; c 3 ) − Ω]+ (1 − E 2 ) 2 tan
2 [Φ ∗ (Φ; c 4 )]
,
(H.71)
r
2 (Λ, Φ; c 3 , c 4 ) =
= A
2
1
cos
2 [Λ
∗ (Λ; c 3 ) − Ω]cos
2 [Φ
∗ (Φ; c 4 )]+ (1 − E
2 )
2 sin
2 [Φ
∗ (Φ; c 4 )]
/
cos
2 [Λ
∗ (Λ; c 3 ) − Ω]cos
2 [Φ
∗ (Φ; c 4 )]+ (1 − E
2 )
2 sin
2 [Φ
∗ (Φ; c 4 )]
3/2
×
×
t
∗
1 (Λ, Φ; c 3 , c 4 ) + t
∗
2 (Λ, Φ; c 3 , c 4 ) + t
∗
3 (Λ, Φ; c 3 , c 4 ) + t
∗
4 (Λ, Φ; c 3 , c 4 )
,
(H.72)
Λ
∗ = c 3 Λ ,
(H.73)
sin Φ
∗ =
= c 4 sin Φ
1 +
2
3
E
2 sin
2 Φ(1 − c
2
4 ) +
1
15
E
4 sin
4 Φ(9 − 20c
2
4 + 11c
4
4 ) + O ( E
6 )
,
(H.74 )
c 1 c 2 c 3 c 4 = 1 .
(H.75 )
End of Lemma.
Corollary H.6 (The equiareal mapping of the biaxial ellipsoid with respect to a transverse frame of reference and a change of scale, special case Ω = 3π/2 (the Hammer projection of E
2
A 1 ,A 2
)).
(i)
The mapping of the right biax ial ellipsoid E
2
A 1 ,A 2
with respect to left biax ial ellipsoid E
2
A 1∗ ,A 2∗
subject
to A 1∗ = A 1 , A 2∗ = A 2 onto the transverse tangent plane specialized by Ω = 3π/2 being normal to
E 3 and with respect to a change of scale is eq uiareal if
x = c 1 r(Λ, Φ; c 3 , c 4 ) cos α(Λ, Φ; c 3 , c 4 ) ,
y = c 2 r(Λ, Φ; c 3 , c 4 ) sin α(Λ, Φ; c 3 , c 4 ) ,
(H.76 )