Generalized Mollweide Projection - Map Projections: Cartographic Information Systems - page 777

G-2 The Generalized Mollweide Projections for the Biaxial Ellipsoid

The characteristics of the spherical Mollweide projection within the class of pseudo-cylindrical
mappings are as follows. The graticule parallel circles are mapped on parallel straight lines , while
meridians on ellipses . We shall keep these properties for the ellipsoidal Mollweide projection by
the “Ansatz”

x ( Λ, Φ )= aΛ cos t, y ( Φ )= b sin t. (G.20)
The polar coordinate t can be interpreted as the reduced latitude of the ellipse x 2 /a 2 ( Λ )+ y 2 /b 2 =1,
where a ( Λ )= aΛ holds. Note that for longitude Λ =0 ,a ( Λ ) = 0 follows. Therefore, the central
Greenwich meridian Λ = 0 is mapped onto a straight line. Of course, any other central meridian
could have been chosen alternatively. Again, we compute the left Cauchy-Green deformation
tensor, this time represented in terms of t ( Φ ) as follows.

∂x
∂Λ = a cos t ( Φ ) ,

∂x
∂Φ =

aΛ sin t ( Φ ) d t

∂y
∂Λ =0 ,

∂y

∂Φ = b cos t ( Φ ) d t

−

d Φ , d Φ ,

(G.21)

( G.13 ),( G.21 )
⇒

⎡ ⎤

a 2 cos 2 t ( Φ ) a 2 Λ sin t ( Φ )cos t ( Φ ) d t

d Φ

−

⎣ ⎦ ,

{ c AB } = (G.22)

−a 2 Λ sin t ( Φ )cos t ( Φ ) d t

d Φ a 2 Λ 2 sin 2 t ( Φ )( d t d Φ ) 2 + b 2 cos 2 t ( Φ )( d t d Φ ) 2

Λ S G AB ( G.14 ), ( G.15 ), ( G.22 ) ⇒| c AB − | =0

⇔

a 2 Λ sin t cos t d t
d Φ

a 2 cos 2 t − Λ S G 11 −

=0

a 2 sin t cos t d t ( a 2 Λ 2 sin 2 t + b 2 cos 2 t )( d t

d Φ ) 2

Λ S G 22 − −

d Φ

⇔

Λ S [ G 11 ( a 2 Λ 2 sin 2 t + b 2 cos 2 t ) d t

2

Λ S G 11 G 22 − + G 22 a 2 cos 2 t ]+

(G.23)

d Φ

+ a 2 cos 2 t ( a 2 Λ 2 sin 2 t + b 2 cos 2 t ) d t
d Φ

2

a 4 Λ 2 sin 2 t cos 2 t d t
d Φ

2

− =0

⇔

Λ S G 11 ( a 2 Λ 2 sin 2 t + b 2 cos 2 t )( d t

d Φ ) 2

+ G 22 a 2 cos 2 t
G 11 G 22

Λ S −

+

G 11 G 22

+ a 2 cos 2 t ( a 2 Λ 2 sin 2 t + b 2 cos 2 t )( d t

a 4 Λ 2 sin 2 t cos 2 t ( d t

d Φ ) 2 d Φ ) 2

− =0 ,

G 11 G 22 G 11 G 22

G 11 ( a 2 Λ 2 sin 2 t + b 2 cos 2 t )( d t

d Φ ) 2

+ G 22 a 2 cos 2 t

Λ S = 1

G 11 G 22 ± 2 G 11 G 22

⎡

G 11 ( a 2 Λ 2 sin 2 t + b 2 cos 2 t )( d t 2

d Φ ) 2

+ G 22 a 2 cos 2 t
G 11 G 22

1
4

⎣

± − (G.24)

G 11 G 22