H-1 The transverse equiareal projection of the biaxial ellipsoid 607
Corollary H.2 (Equiareal mapping of the biaxial ellipsoid onto the transverse tangent plane, special case
Ω = I = 3π/2).
The mapping of the biaxial ellipsoid onto the transverse tangent plane normal to E 3 is equiareal if
(i)
α = A
∗ + π ,
(H.29)
(ii)
r =
A 1 (1 − E
2 )
1/4
(1 − E 2 cos 2 A ∗ ) 3/4 ×
×
− sin B
∗
(1 − E 2 )(1 − E 2 cos 2 A ∗ ) + E 2 cos 2 B ∗ sin
2 A ∗ −
−
1 − 2E
2 cos
2 A
∗ + E
4 cos
2 A
∗
E sin A ∗
arcsin
E sin B
∗ sin A
∗
√
1 − 2E 2 cos 2 A ∗ + E 4 cos 2 A ∗
+
+
(1 − E 2 )(1 − E 2 cos 2 A ∗ )+
+
1 − 2E
2 cos
2 A
∗ + E
4 cos
2 A
∗
E sin A ∗
arcsin
E sin A
∗
√
1 − 2E 2 cos 2 A ∗ + E 4 cos 2 A ∗
1/2
,
(H.30)
(iii)
r = 2A 1 sin
∆
∗
2
=
= A 1
√
2
√
1 − cos ∆ ∗ = A 1
√
2
√
1 − sin B ∗
(if E = 0) .
(H.31)
End of Corollary.
The proof for the integrals is presented in Section H-3.
H-13 The equiareal mapping in terms of ellipsoidal longitude, ellipsoidal latitude
We implement the transformation {Λ
∗ , Φ
∗
} → {A
∗ , B
∗
} into the mapping equations (H.29)–(H.31)
which generate an equiareal mapping onto the transverse tangential plane according to (H.21) and
(H.16), respectively, in particular (H.14). Let us decompose (H.30) term-wise, namely
r
2 (A
∗ , B
∗ ) =
=
A
2
1
√
1 − E 2
(1 − E 2 cos 2 A ∗ ) 3/2 (t 1 + t 2 + t 3 + t 4 ) .
(H.32)
The elaborate computation of the factor and the four terms t 1 , t 2 , t 3 and t 4 as functions of {Λ
∗ , Φ
∗
} has
been performed in Section H-4. Here, the result is presented in form of Lemma H.3 and Corollary H.4.