Transformations between reference systems

Transcription

1 Transformations between reference systems Transformations between reference systems Paolo Zatelli Alfonso Vitti Dept. Civil and Environmental Engineering University of Trento Paolo Zatelli University of Trento 1 / 21

2 Transformations between reference systems Outline 1 Reference systems 2 Transformations between reference systems 3 Transformation between cartographic reference systems Paolo Zatelli University of Trento 2 / 21

3 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 Paolo Zatelli University of Trento 3 / 21

4 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 Paolo Zatelli University of Trento 4 / 21

5 Transformations between reference systems Reference systems Differences between reference systems Paolo Zatelli University of Trento 5 / 21

6 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 Paolo Zatelli University of Trento 6 / 21

7 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 Paolo Zatelli University of Trento 7 / 21

8 Transformations between reference systems Transformations between reference systems Transformations between reference systems 1D In one dimension we fix: origin unit (scale) sense Paolo Zatelli University of Trento 8 / 21

9 Transformations between reference systems Transformations between reference systems Transformations between reference systems 1D The possible transformations can change: the origin x = x + x 0 the scale x = λx the sense x = 1 x, a particular case of scale change with λ = 1 The general transformation is: x = λx + x 0 Paolo Zatelli University of Trento 9 / 21

10 Transformations between reference systems Transformations between reference systems Transformations between reference systems 2D In two dimensions we fix: origin directions scale Paolo Zatelli University of Trento 10 / 21

11 Transformations between reference systems Transformations between reference systems Origin shift y 0 x 0 { x = x + x 0 y = y + y 0 Paolo Zatelli University of Trento 11 / 21

12 Transformations between reference systems Transformations between reference systems Rotation { x = xcos(α) + ysen(α) y = x[ sen(α)] + ycos(α) Paolo Zatelli University of Trento 12 / 21

13 Transformations between reference systems Transformations between reference systems Scale variation { x = λx y = λy Paolo Zatelli University of Trento 13 / 21

14 Transformations between reference systems Transformations between reference systems 7 parameters transformation The general transformation is: { x = λ[xcos(α) + ysen(α)] + x 0 y = λ[x( sen(α)) + ycos(α)] + y 0 in matrix form: [ x y ] [ = λ cos(α) sin(α) sin(α) cos(α) ] [ x y ] [ x0 + y 0 ] Paolo Zatelli University of Trento 14 / 21

15 Transformations between reference systems Transformations between reference systems Transformations between reference systems 3D In three dimensions the transformation is: x x y = λr α,β,γ y + z z x 0 y 0 z 0 with R α,β,γ a rotation matrix of the three angles α, β, γ Paolo Zatelli University of Trento 15 / 21

16 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 Paolo Zatelli University of Trento 16 / 21

17 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 Paolo Zatelli University of Trento 17 / 21

18 Transformations between reference systems Transformation between cartographic reference systems Transformation IGM95 - Rome1940 The transformation used by the IGM is: X Y Z = (1 + K )R x,y,z X Y Z + X 0 Y 0 Z 0 with R x,y,z linearized rotational matrix: 1 R z R y R z 1 R x R y R x 1 Paolo Zatelli University of Trento 18 / 21

19 Transformations between reference systems Transformation between cartographic reference systems Transformation IGM95 - Rome1940 Param. mean s.dev min max X m m Y m m Z m m T m m m m K ppm ppm ppm ppm R x R y R z R Paolo Zatelli University of Trento 19 / 21

20 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 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 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 Surace L., 1998, La georeferenziazione delle informazioni territoriali, Bollettino di geodesia e scienze affini, anno LVII, n. 2, pp Paolo Zatelli University of Trento 20 / 21

21 Transformations between reference systems Appendice Licenza Questa presentazione è c 2009 Paolo Zatelli, disponibile come Paolo Zatelli University of Trento 21 / 21