Coordinates - Map Projections: Cartographic Information Systems - page 145

e 1 0 = e S =+ e 1 sin φ 0 cos λ 0 + e 2 sin φ 0 sin λ 0 − e 3 cos φ 0 ,

e 2 0 = e E = − e 1 sin λ 0 + e 2 cos λ 0 , (3.40)

e 3 0 = e V =+ e 1 cos φ 0 cos λ 0 + e 2 cos φ 0 sin λ 0 + e 3 sin φ 0 .

{ e S , e E , e V | x 0 } is a moving frame (repere mobile) at x 0 := x ( λ 0 ,φ 0 ,r ):
⎡

⎤ ⎡ ⎤ ⎡ ⎤ ⎡ ⎤

e 1 0
e 2 0
e 3 0

e S
e F
e V

sin φ 0 cos λ 0 sin φ 0 sin λ 0 − cos φ 0 e 1

e 2
e 3

⎣ ⎦ = ⎣ ⎦ = ⎣ ⎦ ⎣ ⎦ .

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

− (3.41)

(iv) Parallel transport:

{ e S , e E , e V | x ( λ 0 ,φ 0 ,r ) } = { e s , e E , e V |O} . (3.42)

Statement:

⎡ ⎤ ⎡ ⎤ ⎡ ⎤

e 1 0
e 2 0
e 3 0

e S
e E
e V

e 1
e 2
e 3

⎣ ⎦ = ⎣ ⎦ =R( λ 0 ,φ 0 ,r ) ⎣ ⎦ ,

R( λ 0 ,φ 0 ,r ):=R 2 ( π/ 2 − φ 0 )R 3 ( λ 0 ) , (3.43)

⎡ ⎤ ⎡ ⎤

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

sin φ 0 0 − cos φ 0
01 0
cos φ 0 0 in φ 0

⎣ ⎦ , R 2 ( π/ 2 − φ 0 ):= ⎣ ⎦ .

R 3 ( λ 0 ):=

0 1

S r , equatorial as well as meta-equatorial (oblique) frame of reference

Fig. 3.6. Vertical section of