“Sphere to Cone”: Pseudo-Conic Projections - Map Projections: Cartographic Information Systems - page 443

= 1

4 [( f 2 ) ] 2 h 2 =

= 1

4 [( f 2 ) ] 2 cos 2 Δg 2 ( Δ ) =

(18.16)

= 1

4 [( f 2 ) ] 2 cos 2 Δ 4 R 4 tan 2 Δ

[( f 2 ) ] 2 =

= R 4 sin 2 Δ.

End of Proof.

Proof.

tr[C l G − l ]=
= c 11 G − 1

11 + c 22 G − 1

22 =
= f 2 h 2 ( R 2 sin 2 Δ ) − 1 +( f 2 h 2 Λ 2 + f 2 )( R 2 ) − 1 =

1 1

R 2 sin 2 Δ f 2 g 2 cos 2 Δ + R 2 [ f 2 ( g cos Δ

g sin Δ ) 2 Λ 2 + f 2 ] = = − (18.17)

=4 R 2 f 2

[( f 2 ) ] 2 +

f 2 2 R 2 cos Δ (1 + tan 2 Δ )
( f 2 )

Λ 2 + f 2 ,

2

+ 1
R 2

2 R 2 sin Δ 2 R 2 sin Δ tan Δ

( f 2 ) [( f 2 ) ] 2 ( f 2 ) −

−

det[G l ]= R 4 sin 2 Δ.

(18.18)

End of Proof.

18-2 Special Pseudo-Conic Projections Based Upon the Sphere

The Stab-Werner mapping and the Bonne mapping. The mapping equations and the prin-
cipal stretches. Tissot indicatrix.

We use the setup (“Ansatz”)
r ( Δ )= f ( Δ )= aΔ + b, f ( Δ )= a. (18.19)