Coordinates - Map Projections: Cartographic Information Systems - page 152

P 2 O intersecting the sphere S 2 R , circular meta-equator Fig. 3.8. The oblique plane

Box 3.10 (Establishing an oblique frame of reference (meta-equatorial) of the sphere).
(i) The placement vector X represented in the conventional as well as in the oblique frame
of reference:

X ( Λ, Φ, R )= E 1 R cos Φ cos Λ + E 2 R cos Φ sin Λ + E 3 R sin Φ =

(3.62)

= E 1 R cos B cos A + E 2 R cos B sin A + E 3 R sin B = x ( A, B, R ) .

(ii) The transformation of the frames of reference:

⎡ ⎤ ⎡ ⎤

E 1
E 2
E 3

E 1
E 2
E 3

⎣ ⎦ =R 1 ( I )R 3 ( Ω ) ⎣ ⎦ ,

R 1 ( I )R 3 ( Ω ) = (3.63)

⎡ ⎤

cos Ω sin Ω 0

⎣ ⎦

= − sin Ω cos I cos Ω cos I sin I

sin Ω sin I

− cos Ω sin I cos I

versus

⎡ ⎤ ⎡ ⎤

E 1
E 2
E 3

E 1
E 2
E 3

⎣ ⎦ =R 3 ( Ω )R 1 ( I ) ⎣ ⎦ ,

R 3 ( Ω )R 1 ( I ) =

(3.64)

⎡ ⎤

cos Ω sin Ω cos I sin Ω sin I

sin Ω cos Ω cos I − cos Ω sin I
0

−

⎣ ⎦ .

=

sin I cos I