Mixed Cylindric Map Projections - Map Projections: Cartographic Information Systems - page 767

Fig. F.3. Vertical weighted mean of the generalized Lambert projection and the generalized Sanson-Flamsteed
projection of the biaxial ellipsoid E 2 A,B (generalized Foucaut projection), squared relative eccentricity E 2 =0 . 1,

weight parameters α = β =1

Corollary F.10 (The left eigenvectors of the left Cauchy-Green deformation tensor).

The left eigenvectors of the left Cauchy-Green deformation tensor normalized with respect to the
left metric tensor can be represented with respect to the basis { G 1 , G 2 } which spans the local
tangent space T E

2 A,B

End of Corollary.

The infinitesimal distance dS between two points X and X +d X both elements of the biaxial
ellipsoid

2

E

A,B ,see( F.65 ), is push-forward transformed into the

{ x, y } coordinate representation,

in particular, into ( F.66 )and( F.67 ).

− ( c 12 − Λ 1 G 12 ) G 2 +( c 22 − Λ 1 G 22 ) G 2

f l 1 = G 22 ( c 12 −

Λ 1 G 22 ) ,

(F.65)

Λ 1 G 12 ) 2 + G 11 ( c 22 − Λ 1 G 22 ) 2 − 2 G 12 ( c 12 − Λ 1 G 12 )( c 22 −

− ( c 12 − Λ 2 G 12 ) G 1 +( c 11 − Λ 2 G 11 ) G 2

f l 2 = G 11 ( c 12 − Λ 2 G 12 ) 2 + G 22 ( c 11 − Λ 2 G 11 ) 2

− 2 G 12 ( c 12 − Λ 2 G 12 )( c 11 − Λ 2 G 11 ) ,
d S 2 = G AB d U A d U B = G AB ∂U A
∂u α

∂U B

∂u β d u α d u β

(F.66)

u 1 = x, u 2 = y,
d S 2 = C αβ d u α d u β

∀

C αβ := G AB ∂U A
∂u α

∂U B
∂u β .

∀ (F.67)

The right Jacobi matrix [ ∂U A /∂u α ]=:J r is the inverse of the left Jacobi matrix [ ∂u μ /∂U A ]=:J l ,
i.e. J r =J − l . Since we have already computed J l , we are left with the problem of calculating
J r =J − l =( x Λ y Φ ) − 1 y Φ −

= Λ x Λ y ,

x Φ
0 x Λ

(F.68)

Φ x Φ y

Λ x = x − Λ ,Λ y = x Φ ( x Λ y Φ ) − 1 ,Φ x =0 ,Φ y = y − Φ ,

− (F.69)

C 11 = G 11 Λ x + G 22 Φ x ,
C 12 = G 11 Λ x Λ y + G 22 Φ x Φ y , (F.70)