Transformations between reference systems

Transformations between reference systems
Transformations between reference systems
Paolo Zatelli Alfonso Vitti
Dept. Civil and Environmental Engineering
University of Trento
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Transformations between reference systems
Outline
1
Reference systems
2
Transformations between reference systems
3
Transformation between cartographic reference systems
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Transformations between reference systems | Reference systems
Cartographic systems
The user of cartographic products should know what a reference
system is, that many and different reference systems exist, how to
transform coordinates between systems
The computation of the coordinates transformation between two
system is a task usually performed by a GIS software. The user should
know what “ask” and how to evaluate the software “answer” (results) in
order to avoid gross errors and to estimate the accuracy loss
associated to any transformation
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Transformations between reference systems | Reference systems
Reference systems
One needs to transform coordinates between reference systems
because:
there are global and local systems
there are many local systems
Coordinates may vary of hundreds of meters when a transformation is
applied
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Transformations between reference systems | Reference systems
Differences between reference systems
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Transformations between reference systems | Reference systems
Cartographic systems
To produce a Cartography it is necessary to choose:
the reference system and the associated ellipsoid (Datum)
a set of coordinates and measures that realize the reference
system adopted
the cartographic projection and the related parameters
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Transformations between reference systems | Transformations between reference systems
Transformations between reference systems
We define a reference system by fixing some parameters
corresponding to the degrees of freedom of the system
A transformation “finds” the relation existing between the parameters
of the two systems
It is possible to introduce in the transformation specific functions or
new parameters that can “mitigate” the geometrical distortions of the
reference systems
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Transformations between reference systems | Transformations between reference systems
Transformations between reference systems 1D
In one dimension we fix:
origin
unit (scale)
sense
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Transformations between reference systems | Transformations between reference systems
Transformations between reference systems 1D
The possible transformations can change:
the origin x 0 = x + x0
the scale x 0 = λx
the sense x 0 = −1 ∗ x, a particular case of scale change with
λ = −1
The general transformation is: x 0 = λx + x0
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Transformations between reference systems | Transformations between reference systems
Transformations between reference systems 2D
In two dimensions we fix:
origin
directions
scale
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Transformations between reference systems | Transformations between reference systems
Origin shift
6
6
-
y0
-
x0
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x 0 = x + x0
y 0 = y + y0
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Transformations between reference systems | Transformations between reference systems
Rotation
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x 0 = xcos(α) + ysen(α)
y 0 = x[−sen(α)] + ycos(α)
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Transformations between reference systems | Transformations between reference systems
Scale variation
6
6
-
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-
x 0 = λx
y 0 = λy
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Transformations between reference systems | Transformations between reference systems
7 parameters transformation
The general transformation is:
0
x = λ[xcos(α) + ysen(α)] + x0
y 0 = λ[x(−sen(α)) + ycos(α)] + y0
in matrix form:
0 x
cos(α) sin(α)
x
x0
=λ
+
y0
− sin(α) cos(α)
y
y0
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Transformations between reference systems | Transformations between reference systems
Transformations between reference systems 3D
In three dimensions the transformation is:


 
 0 
x
x0
x
 y 0  = λRα,β,γ  y  +  y0 
z0
z0
z
with Rα,β,γ a rotation matrix of the three angles α, β, γ
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Transformations between reference systems | Transformations between reference systems
Transformation parameters estimate
The parameters are computed using the same expressions used to
transform the coordinates from one reference system to a second
system. The coordinates on both the systems have to be known to
estimate the parameters
In general more than the strictly necessary coordinates are used in the
estimation in order to:
detect gross errors
estimate the standard deviation of the parameters and hence the
accuracy of the transformation
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Transformations between reference systems | Transformation between cartographic reference systems
Cartographic reference systems
The transformation can be applied to geographic coordinates:
(φ, λ, h)1
↓
(φ, λ, h)2
In Cartography we know H and not h.
There are two possible approaches:
1
to compute h from H and the Geoid undulation N as h = N + H
2
to apply a different transformation from (φ, λ, H)1 to (φ, λ, h)2 by
separating planimetry and altimetry, e.g., using the Molodenskij
formula
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Transformations between reference systems | Transformation between cartographic reference systems
Transformation IGM95 - Rome1940
The transformation used by the IGM is:



 
X0
X
X0
 Y 0  = (1 + K )Rx,y ,z  Y  +  Y0 
Z0
Z0
Z

with Rx,y ,z linearized rotational matrix:

1
 −Rz
Ry
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Rz
1
−Rx

−Ry
Rx 
1
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Transformations between reference systems | Transformation between cartographic reference systems
Transformation IGM95 - Rome1940
Param.
X0
Y0
Z0
T
K
Rx
Ry
Rz
R
mean
122.88 m
24.15 m
-3.43 m
144.80 m
18.78 ppm
0.66”
-2.30”
-0.68”
3.45”
Paolo Zatelli | University of Trento |
s.dev
67.43 m
36.27 m
56.35 m
61.32 m
12.90 ppm
1.73”
1.21”
1.75”
1.34”
min
max
41.85 m
-11.39 ppm
552.11 m
54.94 ppm
0.87”
16.58”
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Transformations between reference systems | Appendice | Bibliografia
Bibliografia
Benciolini B., 2004, Dispensa sui sistemi di riferimento, comunicazione
personale.
Di Girolamo A., , Bollettino Ufficiale della Regione Trentino Alto Adige,
n. 19/I-II del 20 aprile 1999.
Donatelli D., Maseroli R., Pierozzi M., 2002, La trasformazione tra i
sistemi di riferimento in Italia, Bollettino di geodesia e scienze affini,
anno LXI, n.4, pp 247–281.
Pierozzi M., Surace L., 2000, I parametri di trasformazione tra il
sistema WGS84 ed il sistema geodetico nazionale Roma40, Bollettino
di geodesia e scienze affini, anno LIX, n.1, pp 37–55.
Surace L., 1998, La georeferenziazione delle informazioni territoriali,
Bollettino di geodesia e scienze affini, anno LVII, n. 2, pp. 181-234.
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Transformations between reference systems | Appendice | Licenza
c
Questa presentazione è 2009
Paolo Zatelli, disponibile come
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