3-3 The oblique frame of reference of the sphere 127
Box 3.3 (Establishing an oblique frame of reference (meta-equatorial) of the sphere).
(i) Placement vector towards the meta-North Pole:
x(λ 0 , φ 0 , r) = e 1 r cos φ 0 cos λ 0 + e 2 r cos φ 0 sin λ 0 + e 3 r sin φ 0 .
(3.37)
(ii) Jacobi map:
D λ 0 x = −e 1 r cos φ 0 sin λ 0 + e 2 r cos φ 0 cos λ 0 ,
D φ 0 x = −e 1 r sin φ 0 cos λ 0 − e 2 r sin φ 0 sin λ 0 + e 3 r cos φ 0 ,
D r x = +e 1 cos φ 0 cos λ 0 + e 2 cos φ 0 sin λ 0 + e 3 sin φ 0 .
(3.38)
(iii) Meta-equatorial (oblique) frame of reference:
˘
South, East, Vertical
¯
:=
:=
j −D φ 0 x
D φ 0 x
,
D λ 0 x
D λ 0 x
,
D r x
D r x
ff
=:
=:
˘
e 1 0 , e 2 0 , e 3 0 [λ 0 , φ 0 , r]
¯
,
(3.39)
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 ,
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 .
(3.40)
{e S , e E , e V x 0 } is a moving frame (rep´ ere mobile) at x 0 := x(λ 0 , φ 0 , r):
2
4
e 1 0
e 2 0
e 3 0
3
5 =
2
4
e S
e E
e V
3
5 =
2
4
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
5
2
4
e 1
e 2
e 3
3
5 .
(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:
2
4
e 1 0
e 2 0
e 3 0
3
5 =
2
4
e S
e E
e V
3
5 = R(λ 0 , φ 0 , r)
2
4
e 1
e 2
e 3
3
5 ,
R(λ 0 , φ 0 , r) := R 2 (π/2 − φ 0 ) R 3 (λ 0 ) ,
R 3 (λ 0 ) :=
2
4
cos λ 0 sin φ 0 0
− sin λ 0 cos λ 0 0
0
0
1
3
5 , R 2 (π/2 − φ 0 ) :=
2
4
sin φ 0 0 − cos φ 0
0
1
0
cos φ 0 0 sinφ 0
3
5 .
(3.43)