Generalized Hammer Projection - Map Projections: Cartographic Information Systems - page 790

x ∗ = r ( A ∗ ,B ∗ )cos A ∗ ,

−

(H.17)

y ∗ = r ( A ∗ ,B ∗ )sin A ∗ . −

Meta-longitude A ∗ coincides with the polar coordinate α , the western azimuth in case of Ω = 270 ◦ ,
in the transverse plane; the radius r is an unknown function r ( A ∗ ,B ∗ ) of meta-longitude A ∗ and
meta-latitude B ∗ which has to be determined. In order to derive the unknown function r ( A ∗ ,B ∗ ),
we calculate the left Cauchy-Green deformation tensor C l := {c KL } , i.e. the infinitesimal distance
between two points in the plane covered by {α,r} ,namely

d s 2 = g kl d u k d u l = g 11 d α 2 + g 22 d r 2 = r 2 d α 2 +d r 2 , (H.18)

∂u k
∂U K

∂ d u l

d s 2 = g kl

∂U L d U K d U L = c KL d U K d U L
= c 11 d A ∗ 2 +2 c 12 d A ∗ d B ∗ + c 22 d B ∗ 2 , (H.19)

c 11 = r 2 ∂α
∂A ∗

2

+ ∂r
∂A ∗

2

,

c 12 = r 2 ∂α
∂A ∗

∂α
∂B ∗ +

∂r
∂A ∗

∂r
∂B ∗ ,

(H.20)

c 22 = r 2 ∂α
∂B ∗

2

+ ∂r
∂B ∗

2

,

∂α
∂A ∗ =1 ,

∂r
∂A ∗ = r A ∗ ,

(H.21)

∂α
∂B ∗

∂r
∂B ∗

=0 , = r B ∗ ,

= r 2 + r A ∗

.

r A ∗ r B ∗

C l := { c KL } (H.22)

r A ∗ r B ∗ r B ∗

Corollary H.1 (Equiareal mapping of the biaxial ellipsoid onto the transverse tangent plane).

The mapping of the biaxial ellipsoid onto the transverse tangent plane normal to E 3 is equiareal
if ( H.23 )and( H.24 ) hold with respect to the left Cauchy-Green deformation tensor C l = { c KL }

2
A 1 ,A 2

of type ( H.22 ) and the left metric tensor G l = { G KL } of E .

det [C l G − l ]=1 , (H.23)

det[ G KL ] .

rr B ∗ = 1

2 r B ∗ = −

(H.24)

End of Corollary.

For the proof of ( H.23 ), we refer to Grafarend ( 1995 ).