Geography Reference
In-Depth Information
Box 21.2 (The conformal group C
7
(3), forward and backward transformation close to the
identity).
R=R
1
(
α
)R
2
(
β
)R
3
(
γ
)
∈
SO(3)
,
(21.6)
⎡
⎤
000
001
0
⎣
⎦
,
R
1
(
α
)=R
1
(0) + R
1
(0)
α
+O
1
(
α
2
)
,
R
1
(0) =
−
10
⎡
⎤
00
−
1
00 0
10 0
⎣
⎦
,
R
2
(
β
)=R
2
(0) + R
2
(0)
β
+O
2
(
β
2
)
,
R
2
(0) =
(21.7)
⎡
⎤
010
−
100
000
⎣
⎦
.
R
3
(
γ
)=R
3
(0) + R
3
(0)
γ
+O
3
(
γ
2
)
,
R
3
(0) =
Forward transformation close to the identity:
x
=
X
+
t
x
−
Zβ
+
Yγ
+
Xs
+O
2
x
,
y
=
Y
+
t
y
+
Zα
−
Xγ
+
Ys
+O
2
y
,
(21.8)
z
=
Z
+
t
z
−
Yα
+
Xβ
+
Zs
+O
2
z
.
Backward transformation close to the identity:
X
=
x
−
t
x
−
yγ
+
zβ
−
xs
+O
2
X
,
Y
=
y
−
t
y
−
zα
+
xγ
−
ys
+O
2
Y
,
(21.9)
Z
=
z
−
t
z
−
xβ
+
yα
−
zs
+O
2
Z
.
“Backward-forward”:
⎡
⎤
t
x
t
y
t
z
α
β
γ
s
⎣
⎦
⎡
⎤
⎡
⎤
x
−
X
100
010
001
0
−
ZY
X
Y
Z
⎣
⎦
=
⎣
⎦
y
−
Y
Z
0
−
X
.
(21.10)
z
−
Z
−
YX
0
Since the “legal” geodetic coordinates relate to longitude
l
and
L
, latitude
b
and
B
,and
height
h
(
l,b
)and
H
(
L, B
) of a topographic point as an element of the Earth's two-dimensional
surface and with respect to an ellipsoid-of-revolution
E
a
1
,a
2
and
E
A
1
,A
2
of local type and global
type, Box
21.3
summarizes the standard transformations of ellipsoidal coordinates
{l,b,h}
and
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