From Riemann Manifolds to Euclidean Manifolds - Map Projections: Cartographic Information Systems - page 124

tr[C l G − l ]

4 .

tr C l G − l 2

Λ 1 , 2 = 1

± − (2.57)

2

Computation of the first invariant tr[C l G − l ]:
tr [C l G − l ]= a 2 cos 2 t

+( t ) 2 a 2 Λ 2 sin 2 t + b 2 cos 2 t
G 22

,

G 11

E 2 cos 2 Φ
(1 − E 2 sin 2 Φ ) 4

π 2

cos 4 t ( t ) 2 =

−E 2 2 ,

ln 1+ E

2 E

−E + 1 1

1

tr[C l G − l ]=

G 11 G 22 [ a 2 G 22 cos 2 t +( t ) 2 G 11 ( a 2 Λ 2 sin 2 t + b 2 cos 2 t )] ,

(2.58)

E 2 sin 2 Φ ) 4
A 1 (1 − E 2 ) 2 cos 2 Φ ,

1
G 11 G 22 =

(1 −

A 1 cos 2 Φ

G 11 =

E 2 sin 2 Φ , 1 −

A 1 (1 E 2 ) 2

−

G 22 =

E 2 sin 2 Φ ) 3 . (1 −

Box 2.5 (Left Cauchy-Green matrix, generalized Mollweide projection of the ellipsoid-of-
revolution).

Left Jacobi matrix:
J l := D Λ xD Φ x

,

D Λ yD Φ y

(2.59)

aΛ sin tt ,
D Λy =0 , D Φy = D t yD Φ t =+ b cos tt .
Left Cauchy-Green matrix:
C l := J l G r J l , G r =I 2

D Λ x = a cos t, D Φ x = D t xD Φ t = −

C l =J l J l , ⇒

C l =

aΛ 2 cos t sin tt ( a 2 Λ 2 sin 2 t + b 2 cos 2 t )( t ) 2 .

a 2 cos 2 t aΛ 2 cos t sin tt

−

(2.60)

−

Left matrix of the metric:
G l = N 2 ( Φ )0 ( N ( Φ )and M ( Φ ): seeExample 1.3 ) . (2.61)

M 2 ( Φ )

0

det [C l G − l ]=1:

a 2 Λ 2 sin 2 t + b 2 cos 2 t
G 22

a 4 Λ 2 cos 2 t sin 2 t
G 11 G 22

det [C l G l 1] = a 2 cos 2 t ( t ) 2 ( t ) 2 =

− (2.62)

G 11

cos 4 t

G 11 G 22 a 2 b 2 ( t ) 2 .

=