Coordinates - Map Projections: Cartographic Information Systems - page 173

Y = − X sin Ω cos I + Y cos Ω cos I + Z sin I =

A 1

E 2 )sin Φ sin I )=

= 1 − E 2 sin 2 Φ ( − cos Φ cos Λ sin Ω cos I +cos Φ sin Λ cos Ω cos I +(1 −

(3.145)

A 1

1

E 2 )sin Φ sin I ) , = (+ cos Φ cos I sin( Λ − Ω )+(1 −

E 2 sin 2 Φ

−

Z = X sin Ω sin I − Y cos Ω cos I + Z cos I =

A 1

E 2 )sin Φ cos I ) , (3.146)

1

= (cos Φ cos Λ sin Ω sin I − cos Φ sin Λ cos Ω cos I +(1 −

E 2 sin 2 Φ −

such that
tan A = − cos Φ cos I sin( Λ − Ω )+(1 − E 2 )sin Φ sin I
cos Φ cos( Λ

,

− Ω )

E 2 )sin Φ cos I

cos Φ cos Λ sin Ω sin I − cos Φ sin Λ cos Ω cos I +(1 −

tan B = cos 2 Φ cos 2 ( Λ

E 2 )sin Φ sin I ] 2 .

(3.147)

− Ω )+[cos Φ cos I sin( Λ − Ω )+(1 −

Let us here additionally collect the result of the transformation { Λ, Φ }→{ A, B } by the following

Corollary 3.10 .

Corollary 3.10 (The change from one chart to another chart: cha-cha-cha, the surface normal
ellipsoidal longitude/latitude versus the oblique quasi-spherical longitude/latitude).

Given the longitude of the ascending node Ω as well as the inclination I of the oblique equatorial
plane, then the transformation of surface normal ellipsoidal longitude/latitude into oblique quasi-
spherical longitude/latitude is represented by ( 3.147 ).

End of Corollary.

of type oblique quasi-spherical longitude/latitude
are not orthogonal. Accordingly, the matrix of the metric of E A 1 ,A 2 in terms of these coordinates
contains off-diagonal elements. Finally, we note that the terms up to order three of the corre-
sponding Taylor series expansions can be determined by resorting to the partial derivatives of
Box 3.23 .

It should be noted that the coordinates { A, B }

Box 3.23 (Partial derivatives up to order three).

cos 2 B cos I − sin A sin B cos B sin I
(cos A cos B cos Ω

(tan Λ ) ,A =+

sin A cos B sin Ω cos I +sin B sin Ω sin I ) 2 , −

(3.148)

cos A sin I

(tan Λ ) ,B = −

sin A cos B sin Ω cos I +sin B sin Ω sin I ) 2 ; (cos A cos B cos Ω −

N = N ( A, B ):=cos 2 B cos I

− sin A sin B cos B sin I,

M = M ( A, B ):=cos A sin I, (3.149)