Geography Reference
In-Depth Information
Fig. 21.6.
Polynomial diagram, the polynomial representation of the conformal coordinate Easting
X
in a global
frame of reference of type Gauss-Krueger or UTM, the
solid dots
illustrate non-zero monomials, the
open circles
zero monomials, according to
Cox et al.
(
1996
, pp. 433-443)
21-412 The Second Step: Curvilinear Datum Transformation
With reference to
Grafarend and Syffus
(
1995
, pp. 344-348), let us introduce the curvilinear
datum transformation from a local datum (regional, National, European) to a global datum
close to the identity reviewed in Box
21.12
.Notethat
l
and
b
refer to surface normal ellipsoidal
longitude/ellipsoidal latitude of an ellipsoid-of-revolution
in a local frame of reference (semi-
major axis
a
1
, semi-minor axis
a
2
, relative eccentricity squared
e
2
:= (
a
1
−
E
a
1
,a
2
a
2
)
/a
1
). The datum
transformation close to the identity is expressed as a linear function of the datum transformation
parameters, three parameters of translation
and
one scale parameter
s
. Note that the datum transformation in longitude close to the identity is
independent of the translation parameter
t
z
and of the scale parameter
s
. In contrast, the datum
transformation in latitude close to the identity does not depend on the rotation parameter
γ
(
z
axis rotation). The coecients
{
t
x
,t
y
,t
z
}
, three parameters of rotation
{
α,β,γ
}
are given in Box
21.13
. The synthesis matrix S of a
datum transformation close to the identity depends on the ellipsoidal height
h
in a local frame of
reference. Only in a few cases local ellipsoidal height information is available. Within the matrix
decomposition S(
h
)=S
0
+
h
S
1
+O(
h
2
), we study the influence of local ellipsoidal height
h
.
{
s
10
, ..., s
27
}
Box 21.12 (Curvilinear datum transformation, synthesis close to the identity).
L
=
=
l
+
s
10
+
s
11
t
x
+
s
12
t
y
+
s
13
t
z
+
s
14
α
+
s
15
β
+
s
16
γ
+
s
17
s
=
l
+
δL,
δL
;=
=
s
10
+
s
11
t
x
+
s
12
t
y
+
s
13
t
z
+
s
14
α
+
s
15
β
+
s
16
γ
+
s
17
s,
(21.67)
B
=
=
b
+
s
20
+
s
21
t
x
+
s
22
t
y
+
s
23
t
z
+
s
24
α
+
s
25
β
+
s
26
γ
+
s
27
s
=
b
+
δB,
δB
:=
=
s
20
+
s
21
t
x
+
s
22
t
y
+
s
23
t
z
+
s
24
α
+
s
25
β
+
s
26
γ
+
s
27
s.
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