“Sphere to Cone”: Pseudo-Conic Projections - Map Projections: Cartographic Information Systems - page 442

Right metric tensor:
G r = r 2 0

= f 2 0

(18.10)

01 01

Left metric tensor:
G l = R 2 sin 2 Δ 0

R 2 , det[G l ]= R 4 sin 2 Δ.

(18.11)

0

Postulate of an equal area mapping:

det[C l ]=det[G l ] ⇒ ff g cos Δ =+ R 2 sin Δ
(only the + sign is here correct due to the orientation constance) (18.12)

⇔

g =2 R 2 tan Δ
( f 2 )

= R 2 tan Δ
ff

.

(18.13)

General structure
(equal area mapping: pseudo-conic) :

α = α ( Λ, Δ )= g ( Δ ) Λ cos Δ =2 R 2 sin Δ

[ f 2 ( Δ )] Λ = R 2 sin Δ

ff Λ,

(18.14)

r = r ( Δ )= f ( Δ ) .

Proof.

( Λ S ) + =

tr[C l G − l ]+ (tr[C l G − l ]) 2

− 4det[C l G − l ] ,

= 1
2

(18.15)

( Λ S ) − =

tr [C l G − l ]

4det[C l G − l ] ,

(tr[C l G − l ]) 2

= 1
2

− −

det[C l ]=