“Ellipsoid-of-Revolution to Cylinder”: Oblique Aspect - Map Projections: Cartographic Information Systems - page 418

× [ A 1 (1 − E 2 ) 2 + E 2 A 2 sin 2 i sin 2 α 0 ][ A 1 +( A 2 cos 2 i − A 1 )sin 2 α 0 ]sin α 0 +
+2[ A 1 +( A 2 cos 2 i A 1 )sin 2 α 0 ] E 2 A 2 sin 2 i sin α 0 cos 2 α 0 +

−

A 1 )sin α 0 cos 2 α 0 .

+[ A 1 (1 E 2 ) 2 + E 2 A 2 sin 2 i sin 2 α 0 ]( A 2 cos 2 i

− −

Box 16.5 (Longitude L ( α ) as a function of meta-longitude α ).
Power series expansion Δl = N

r =1 l r Δα r up to order N =2:

A 1 A 2 cos i

l 1 := +

A 1 cos 2 α 0 A 2 cos 2 i sin 2 α 0 ,

(16.58)

2 A 1 A 2 cos i ( A 2 cos 2 i

A 1 )sin α 0 cos α 0
( A 1 cos 2 α 0 + A 2 cos 2 i sin 2 α 0 ) 2

−

2 l 2 := − .

The relations ( 16.44 ) together with the relations ( 16.50 ) lead to the relations ( 16.51 ). Let us prove
the other central relations here.

Proof (( 16.47 ): b 1 ).

A 2

sin α
A 1 +( A 2 cos 2 i tan B =

1 − E 2 sin i

A 1 )sin 2 α ,

−

A 1 cos α
[ A 1 +( A 1 cos 2 i

A 2 dtan B

d α

1
cos 2 B

d B
d α =

=

E 2 sin i

A 1 )sin 2 α ] 3 / 2 ,

1 −

−

d B

d α =cos 2 B dtan B

1
1+tan 2 B

dtan B
d α

= = (16.59)

d α

=[ A 2 A 1 (1 E 2 )sin i cos α ][ A 1 +( A 2 cos 2 i A 1 )sin 2 α ] − 1 / 2 − − ×

× [ A 1 (1 − E 2 ) 2 + E 2 A 2 sin 2 i sin 2 α ] − 1
⇒
b 1 := d B

d α ( α 0 ) .

End of Proof.