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Corollary 24.1 (Ten parameter conformal group
C
(3) close to the identity).
Let
X
=[
X
1
,X
2
,X
3
]
,
Y
=[
Y
1
,Y
2
,Y
2
],
respectively represent the coordinates x
μ
,
x
μ
,
respec-
tively. Then in terms of the ten transformation parameters of type translation, rotation, special
conformal transformation, dilatation, namely the partitioned transformation parameter vector.
x
10
]
T
=
X
=[
x
1
,x
2
,x
3
|
x
4
,x
5
,x
6
|
x
7
,x
8
,x
9
|
[
δα
1
,δα
2
,δα
3
δω
1
,δω
1
,δω
2
δc
1
,δc
2
,δc
3
|δλ
]
T
,
the conformal group
C
(3) close to the identity can be respected by
⎡
⎤
⎡
⎤
⎡
⎤
⎡
⎤
100
010
001
x
1
x
2
x
3
−
X
2
−
X
3
0
x
4
x
5
x
6
⎣
⎦
⎣
⎦
+
⎣
⎦
⎣
⎦
+
Y
−
X
=
X
1
0
−
X
3
0
X
1
X
2
⎡
⎤
⎡
⎤
⎡
⎤
X
1
+
X
2
+
X
3
−
−
2
X
1
X
2
−
2
X
1
X
3
x
7
x
8
x
9
x
1
x
2
x
3
⎣
⎦
⎣
⎦
+
⎣
⎦
x
10
X
1
−
X
2
+
X
3
−
2
X
1
X
2
−
2
X
2
X
3
(24.6)
X
1
+
X
2
−
X
3
−
2
X
1
X
3
−
2
X
2
X
3
or
⎡
⎤
x
1
x
2
x
3
x
4
x
5
x
6
x
7
x
8
x
9
x
10
⎣
⎦
⎡
⎤
X
1
+
X
2
+
X
3
100
010
001
−
X
2
−
X
3
0
−
−
2
X
1
X
2
−
2
X
1
X
3
X
1
X
2
X
3
⎣
⎦
X
1
−
X
2
+
X
3
Y
−
X
=
X
1
0
−
X
3
−
2
X
1
X
2
−
2
X
2
X
3
X
1
+
X
2
−
X
3
0
X
1
X
2
−
2
X
1
X
3
−
2
X
2
X
3
(24.7)
End of Corollary.
24-2 The Ten Parameter Determination of the Conformal Group
in Three-Dimensional Euclidean Space, Numerical Examples
From two well-known data sets of three-dimensional Cartesian coordinates
Heindl 1986
)weare
going to determine the ten parameters of
C
10
(3) close to the identity. Tables
24.1
and
24.2
lists
twice 45 Cartesian coordinates of 15 points. The following computations have been carried out in
order to illustrate the benefits of the more general datum transformation
C
10
(3). Basically the
transformation has been applied to the original data
without reduction
to the “
centre of gravity
”
as well as to those data with reduction to the “
centre of gravity
”. Additionally we have studied
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