21-1 Analysis of a datum problem 455
Since the datum parameters of a geodetic datum transformation are usually close to the identity,
Box 21.2 is a collection of the rotation matrix close to the identity, parameterized by Cardan angles
into the forward and backward transformations (21.8) and (21.9) of that type, finally by (21.10) given
as a system of linear equations with seven parameters (three for translation, three for rotation, and
one for scale) as unknowns.
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)
R 1 (α) = R 1 (0) + R
1 (0)α + O 1 (α
2 ) ,
R
1 (0) =
2
6
4
0 0 0
0 0 1
0 −1 0
3
7
5 ,
R 2 (β) = R 2 (0) + R
2 (0)β + O 2 (β
2 ) ,
R
2 (0) =
2
6
4
0 0 −1
0 0 0
1 0 0
3
7
5 ,
R 3 (γ) = R 3 (0) + R
3 (0)γ + O 3 (γ
2 ) ,
R
3 (0) =
2
6
4
0 1 0
−1 0 0
0 0 0
3
7
5 .
(21.7)
Forward transformation close to the identity:
x = X + t x − Zβ + Y γ + Xs + O 2x ,
y = Y + t y + Zα − Xγ + Y s + O 2y ,
z = Z + t z − Y α + Xβ + Zs + O 2z .
(21.8)
Backward transformation close to the identity:
X = x − t x − yγ + zβ − xs + O 2X ,
Y = y − t y − zα + xγ − ys + O 2Y ,
Z = z − t z − xβ + yα − zs + O 2Z .
(21.9)
“Backward–forward”:
2
6
4
x − X
y − Y
z − Z
3
7
5 =
2
6
4
1 0 0 | 0 −Z Y |X
|
|
|
|
0 1 0 | Z
0 −X |Y
|
|
|
|
0 0 1 |−Y X
0 |Z
3
7
5
2
6
6
6
6
6
6
6
6
6
6
6
6
6
4
t x
t y
t z
α
β
γ
s
3
7
7
7
7
7
7
7
7
7
7
7
7
7
5
.
(21.10)