H-2 The ellipsoidal Hammer projection 609
Corollary H.4 (Equiareal mapping of the biaxial ellipsoid onto the transverse tangent plane, special case
Ω = 3π/2).
The mapping of the biaxial ellipsoid onto the transverse tangent plane normal to E 3 is equiareal if
(i)
α = arctan
(1 − E
2 ) tan Φ
∗
sin Λ ∗
,
r = A 1
sin
2 Λ ∗ cos 2 Φ ∗ + (1 − E 2 ) 2 sin
2 Φ ∗
(sin
2 Λ ∗ cos 2 Φ ∗ + (1 − E 2 ) sin
2 Φ ∗ ) 3/4
t ∗
1 + t ∗
2 + t ∗
3 + t ∗
4
(in polar coordinates) ,
(H.37)
subject to
t
∗
1 = −
cos Λ
∗ cos Φ
∗
1 − E 2 sin
2 Φ ∗
sin
2 Λ ∗ cos 2 Φ ∗ + (1 − E 2 ) sin
2 Φ ∗ ,
t
∗
2 = −
1 − cos
2 Λ
∗ cos
2 Φ
∗
E sin Φ ∗
arcsin
E cos Λ
∗ sin Φ
∗ cos Φ
∗
(1 − E 2 sin
2 Φ ∗ )(1 − cos 2 Λ ∗ cos 2 Φ ∗ )
,
t
∗
3 =
sin
2 Λ ∗ cos 2 Φ ∗ + (1 − E 2 ) sin
2 Φ ∗ ,
t
∗
4 =
1 − cos
2 Λ
∗ cos
2 Φ
∗
E sin Φ ∗
arcsin
E sin Φ
∗
√
1 − cos 2 Λ ∗ cos 2 Φ ∗
,
(H.38)
(ii)
cos α =
sin Λ
∗
sin
2 Λ ∗ + (1 − E 2 ) 2 tan
2 Φ ∗
, sin α =
(1 − E
2 ) tan Φ
∗
sin
2 Λ ∗ + (1 − E 2 ) 2 tan
2 Φ ∗
(in Cartesian coordinates x
∗ = r cos α and y
∗ = r sin α) ,
(H.39)
(iii)
α = lim
E→0
α(E) = arctan
tan Φ
∗
sin Λ ∗ , r = lim
E→0
r(E) = A 1
√
2
√
1 − cos Λ ∗ cos Φ ∗
(if E = 0) .
(H.40)
End of Corollary.
H-2 The ellipsoidal Hammer projection
The second constituent of the Hammer projection is a proper change of scale of the transverse equiareal
projection which conserves the local area.