Generalized Mollweide Projection - Map Projections: Cartographic Information Systems - page 779

1

2 t + 1

4 sin 2 t =

(G.30)

2 ar tanh( E sin Φ ) .

= A 1
ab

1 − E 2
E

E sin Φ

E 2 sin 2 Φ ) + 1 2(1 −

Due to ar tanh( E sin Φ )= 2 ln 1+ E sin Φ

1 −E sin Φ , we alternatively obtain ( G.31 ).

2 t + sin 2 t =

ln 1+ E sin Φ
1

.

= A 1
ab

1 − E 2
E

2 E sin Φ

E sin Φ +

(G.31)

E 2 sin 2 Φ − 1 −

2
A 1 ,A 2

Note that the total area of the biaxial ellipsoid (for example, Grafarend 1992a )isrep-

resented by ( G.32 ). This result can be used to determine the unknown ellipsoid axes a and b
according to the Mollweide gauge . In case of the sphere

E

2 R . Mollweide ( 1805 ) has proposed that
the half-sphere −π|/ 2 ≤ Λ ≤ + π/ 2 should be mapped equiareally onto a circle in the plane
P

S

, a corresponding postulate would define the half-ellipsoid
−π/ 2 ≤ Λ ≤ + π/ 2 to be mapped equiareally onto an ellipse , in particular ( G.33 ).

2 . For the biaxial ellipsoid E A 1 ,A 2

ln 1+ E
1 E 2 ,

E 2
E

= πA 1 1

− 2 E

S E 2 A 1 ,A 2

E +

(G.32)

− 1 −

a ( Λ = π/ 2) = aπ/ 2 ,πa ( Λ = π/ 2) b = 1

2 S E 2 A 1 ,A 2

⇔

1

2 π 2 ab = 1 2 S E 2 A 1 ,A 2

(G.33)

⇔
π 2 ab = S E 2 A 1 ,A 2
⇔

ln 1+ E
1 E 2 .

πab = A 1 1 − E 2

2 E

E +

(G.34)

E − 1 −

Corollary G.3 (Generalized Mollweide gauge for the biaxial ellipsoid).

Under the postulate that the half-ellipsoid −π/ 2 ≤ Λ ≤ + π/ 2 to be mapped equiareally onto
an ellipse in P

2 , the generalized Mollweide gauge ( G.34 ) holds. The result ( G.34 ) approaches the
original Mollweide gauge once we set the relative eccentricity E =0.

End of Corollary.