Coordinates - Map Projections: Cartographic Information Systems - page 168

subject to

cos 2 I sin 2 A ) 2 +cos 4 I sin 2 A cos 2 A +
+ E 6 (1 − cos 2 I sin 2 A ) 2 +3cos 4 I (1 − cos 2 I sin 2 A )sin 2 A cos 2 A +O + ( E 8 );

cos 2 I sin 2 A )+ E 4 (1

x := E 2 (1

− −

(3.124)

E 2 1+ 1

d S ( B =0)= A 1 √ 1

2 E 2 (1

cos 2 I sin 2 A )+

− −

+ E 4 3 cos 2 I sin 2 A ) 2 +cos 4 I sin 2 A cos 2 A +

8 (1

− (3.125)

+ E 6 5 cos 2 I sin 2 A )sin 2 A cos 2 A cos 2 I sin 2 A ) 3 + 5

4 cos 4 I (1 16 (1

− −

+O + ( E 8 ) d A.

End of Solution (the fourth step).

Note that all series ( 3.120 ), ( 3.121 ), ( 3.122 ), and ( 3.125 ) are uniformly convergent. Accordingly,
we can interchange integration and summation within ( 3.117 ) when we substitute ( 3.125 )asa
series expansion. An alternative useful expansion of S ( A ) in terms of powers of ΔA is provided
by the following formulae:

S ( A 0 + ΔA )= S ( A 0 )+ S 1 ( A 0 ) ΔA + S 2 ( A 0 )( ΔA ) 2 +O S [( ΔA ) 3 ] ,

(3.126)

E 2 1+ 1

d A = A 1 √ 1

S 1 ( A )= 1
1!

d S

2 E 2 (1

cos 2 I sin 2 A )+ − −

+ E 4 3

8 (1 − cos 2 I sin 2 A ) 2 +cos 4 I sin 2 A cos 2 A +

(3.127)

+ E 6 5 cos 2 I sin 2 A )sin 2 A cos 2 A +O( E 8 ) ,

cos 2 I sin 2 A ) 3 + 5

4 cos 4 I (1

16 (1

− −

d A 2 = A 1 √ 1

E 2 −

d 2 S

S 2 ( A )= 1
2!

E 2 cos 2 I sin A cos A

− −

−E 4 3

4 (1 − cos 2 I sin 2 A )cos 2 I sin A cos A − 2cos 2 I sin A cos 3 A +cos 4 I sin 3 A cos A

−

(3.128)

E 6 15

cos 2 I sin 2 A ) 2 cos 2 I sin A cos A + 5

2 cos 2 I sin 3 A cos A

−

8 (1

− −

cos 2 I sin 2 A )cos 4 I sin 3 A cos A +O( E 8 ) .

5
2 (1

cos 2 I sin 2 A )cos 4 I sin A cos 3 A + 5

− −

4 (1

−

Next, we have to work out how the oblique quasi-spherical longitude/latitude are related to the
standard surface normal ellipsoidal longitude/latitude. This finally concludes the introduction of
the oblique coordinate system of the ellipsoid-of-revolution. At first, we here aim at a transforma-
tion of oblique quasi-spherical longitude/latitude into surface normal ellipsoidal longitude/latitude
to which we refer as direct transformation . Additionally, we here aim at a transformation of sur-
face normal ellipsoidal longitude/latitude into oblique quasi-spherical longitude/latitude to which
we refer as inverse transformation .