Coordinates - Map Projections: Cartographic Information Systems - page 160

⎡ ⎤ ⎡ ⎤

E 1
E 2
E 3

E 1
E 2
E 3

⎣ ⎦ =R 1 ( I )R 3 ( Ω ) ⎣ ⎦ ,

⎡ ⎤ ⎡ ⎤

E 1
E 2
E 3

E 1
E 2
E 3

⎣ ⎦ =R 3 ( ω )R 1 ( I )R 3 ( Ω ) ⎣ ⎦ .

(3.92)

(ii) e → e 0

⎡ ⎤

⎦ =R 2 π

⎡ ⎤

e 1 0
e 2 0
e 3 0

e 1
e 2
e 3

φ 0 R 3 ( λ 0 )

⎣ ⎣ ⎦ .

2 −

(3.93)

(iii) E = e, E = e 0

⎡ ⎤ ⎡ ⎤

E 1
E 2
E 3

E 1
E 2
E 3

⎣ ⎦ =R 3 ( ω )R 1 ( I )R 3 ( Ω ) ⎣ ⎦

⎡ ⎤ ⎡ ⎤

e 1
e 2
e 3

e 1 0
e 2 0
e 3 0

=R 2 π Φ 0 R 3 ( λ 0 )

⎣ ⎦ = ⎣ ⎦

2 −

⇔

(3.94)

R 3 ( ω )R 1 ( I )R 3 ( Ω )=R 2 π

2 − φ 0 R 3 ( λ 0 ) .

Box 3.19 (Individual equations of transforming oblique frames of reference).

Equation (i):
R 2 ( ω )R 1 ( I )R 3 ( Ω )=

⎡ ⎤

cos ω cos Ω − sin ω sin Ω cos I cos ω sin Ω +sin ω cos Ω cos I sin ω sin I

⎣ ⎦ .

= − sin ω cos Ω − cos ω sin Ω cos I − sin ω sin Ω +cos ω cos Ω cos I cos ω sin I (3.95)

sin Ω sin I − cos Ω sin I cos I

Equation (ii) :
R 2 π φ 0 R 3 ( λ 0 )=

2 −

⎡ ⎤

sin φ 0 cos λ 0 sin φ 0 sin λ 0 − cos φ 0

⎣ ⎦ .

= sin λ 0 cos λ 0 0

cos φ 0 cos λ 0 cos φ 0 sin λ 0 sin φ 0

− (3.96)