130 3 Coordinates
Box 3.4 (Representation of a placement vector x ∈ S
2
r in both the equatorial frame of reference indicated
by {e 1 , e 2 , e 3 O} and the meta-equatorial (oblique) frame of reference indicated by {e 1 0 , e 2 0 , e 3 0 O}).
e 1 x + e 2 y + e 3 z = x = e 1 0 x
0 + e 2 0 y
0 + e 3 0 z
0 ,
e 1 cos φ cos λ + e 2 cos φ sin λ + e 3 sin φ =
x
r
= e 1 0 cos β cos α + e 2 0 cos β sin α + e 3 0 sin β .
(3.44)
Transformation {e 1 , e 2 , e 3 O} → {e 1 0 , e 2 0 , e 3 0 O}:
2
4
e 1
e 2
e 3
3
5 =
2
4
sin φ 0 cos λ 0 − sin λ 0 cos φ 0 cos λ 0
sin φ 0 sin λ 0 cos λ 0 cos φ 0 sin λ 0
− cos φ 0
0
s i nφ 0
3
5
2
4
e 1 0
e 2 0
e 3 0
3
5 ,
(3.45)
e 1 = +e 1 0 sin φ 0 cos λ 0 − e 2 0 sin λ 0 + e 3 0 cos φ 0 cos λ 0 ,
e 2 = +e 1 0 sin φ 0 sin λ 0 + e 2 0 cos λ 0 + e 3 0 cos φ 0 sin λ 0 ,
e 3 = −e 1 0 cos φ 0 + e 3 0 sin φ 0 ,
(3.46)
x = e 1 x + e 2 y + e 3 z =
= e 1 0
`
x sin φ 0 cos λ 0 + y sin φ 0 sin λ 0 − z cos φ 0
´
+
+e 2 0
` − x sin λ 0 + y cos λ 0
´
+ e 3 0
`
x cos φ 0 cos λ 0 + y cos φ 0 sin λ 0 + z sin φ 0
´
=
= re 1 0
`
cos φ cos λ sin φ 0 cos λ 0 + cos φ sin λ sin φ 0 sin λ 0 − sin φ cos φ 0
´
+
+re 2 0
` − cos φ cos λ sin λ 0 + cos φ sin λ cos λ 0
´
+
+re 3 0
`
cos φ cos λ cos φ 0 cos λ 0 + cos φ sin λ cos φ 0 sin λ 0 + sin φ sin φ 0
´
= re 1 0 cos β cos α + re 2 0 cos β sin α + re 3 0 sin β .
(3.47)
Box 3.5 (From the equatorial frame of reference {e 1 , e 2 , e 3 O} to the meta-equatorial (oblique) frame of
reference {e 1 0 , e 2 0 , e 3 0 O}: the direct transformation).
(i) The first identity (spherical sine lemma):
x
0 = r cos β cos α ⇒ cos α =
x
0
r cos β
,
cos α =
1
cos β
`
cos φ cos λ sin φ 0 cos λ 0 + cos φ sin λ sin φ 0 sin λ 0 − sin φ cos φ 0
´
,
cos α =
1
cos β
`
cos φ sin φ 0 cos(λ − λ 0 ) − sin φ cos φ 0
´
.
(3.48)
(ii) The second identity (spherical sine-cosine lemma):
y
0 = r cos β sin α ⇒ sin α =
y
0
r cos β
,
sin α =
1
cos β
` − cos φ cos λ sin λ 0 + cos φ sin λ cos λ 0
´
,
sin α =
1
cos β
cos φ sin(λ − λ 0 ) .
(3.49)
(iii) The third identity (spherical side cosine lemma):
z
0 = r sin β = r cos ψ ⇒ sin β = cos ψ =
z
0
r
,
sin β = sin ψ = cos φ cos φ 0 cos λ cos λ 0 + cos φ cos φ 0 sin λ sin λ 0 + sin φ sin φ 0 ,
sin β = sin ψ = cos φ cos φ 0 cos(λ − λ 0 ) + sin φ sin φ 0 .
(3.50)
Box 3.4 (Representation of a placement vector x ∈ S
2
r in both the equatorial frame of reference indicated
by {e 1 , e 2 , e 3 O} and the meta-equatorial (oblique) frame of reference indicated by {e 1 0 , e 2 0 , e 3 0 O}).
e 1 x + e 2 y + e 3 z = x = e 1 0 x
0 + e 2 0 y
0 + e 3 0 z
0 ,
e 1 cos φ cos λ + e 2 cos φ sin λ + e 3 sin φ =
x
r
= e 1 0 cos β cos α + e 2 0 cos β sin α + e 3 0 sin β .
(3.44)
Transformation {e 1 , e 2 , e 3 O} → {e 1 0 , e 2 0 , e 3 0 O}:
2
4
e 1
e 2
e 3
3
5 =
2
4
sin φ 0 cos λ 0 − sin λ 0 cos φ 0 cos λ 0
sin φ 0 sin λ 0 cos λ 0 cos φ 0 sin λ 0
− cos φ 0
0
s i nφ 0
3
5
2
4
e 1 0
e 2 0
e 3 0
3
5 ,
(3.45)
e 1 = +e 1 0 sin φ 0 cos λ 0 − e 2 0 sin λ 0 + e 3 0 cos φ 0 cos λ 0 ,
e 2 = +e 1 0 sin φ 0 sin λ 0 + e 2 0 cos λ 0 + e 3 0 cos φ 0 sin λ 0 ,
e 3 = −e 1 0 cos φ 0 + e 3 0 sin φ 0 ,
(3.46)
x = e 1 x + e 2 y + e 3 z =
= e 1 0
`
x sin φ 0 cos λ 0 + y sin φ 0 sin λ 0 − z cos φ 0
´
+
+e 2 0
` − x sin λ 0 + y cos λ 0
´
+ e 3 0
`
x cos φ 0 cos λ 0 + y cos φ 0 sin λ 0 + z sin φ 0
´
=
= re 1 0
`
cos φ cos λ sin φ 0 cos λ 0 + cos φ sin λ sin φ 0 sin λ 0 − sin φ cos φ 0
´
+
+re 2 0
` − cos φ cos λ sin λ 0 + cos φ sin λ cos λ 0
´
+
+re 3 0
`
cos φ cos λ cos φ 0 cos λ 0 + cos φ sin λ cos φ 0 sin λ 0 + sin φ sin φ 0
´
= re 1 0 cos β cos α + re 2 0 cos β sin α + re 3 0 sin β .
(3.47)
Box 3.5 (From the equatorial frame of reference {e 1 , e 2 , e 3 O} to the meta-equatorial (oblique) frame of
reference {e 1 0 , e 2 0 , e 3 0 O}: the direct transformation).
(i) The first identity (spherical sine lemma):
x
0 = r cos β cos α ⇒ cos α =
x
0
r cos β
,
cos α =
1
cos β
`
cos φ cos λ sin φ 0 cos λ 0 + cos φ sin λ sin φ 0 sin λ 0 − sin φ cos φ 0
´
,
cos α =
1
cos β
`
cos φ sin φ 0 cos(λ − λ 0 ) − sin φ cos φ 0
´
.
(3.48)
(ii) The second identity (spherical sine-cosine lemma):
y
0 = r cos β sin α ⇒ sin α =
y
0
r cos β
,
sin α =
1
cos β
` − cos φ cos λ sin λ 0 + cos φ sin λ cos λ 0
´
,
sin α =
1
cos β
cos φ sin(λ − λ 0 ) .
(3.49)
(iii) The third identity (spherical side cosine lemma):
z
0 = r sin β = r cos ψ ⇒ sin β = cos ψ =
z
0
r
,
sin β = sin ψ = cos φ cos φ 0 cos λ cos λ 0 + cos φ cos φ 0 sin λ sin λ 0 + sin φ sin φ 0 ,
sin β = sin ψ = cos φ cos φ 0 cos(λ − λ 0 ) + sin φ sin φ 0 .
(3.50)