Fig.: Global geozentrisches Koordinatensystem
Source: Bauer, 2011
MGE-01, Transforming to Official Coordinates, Holst/Medic 3 WS 20/21
β’ Similarity Transformation between coordinate systems:
πβ²πβ²πβ²
= ππΉ3(ππ§)πΉ2(ππ¦)πΉ1(ππ₯)
πΉ
πππ
+
π‘π₯π‘π¦π‘π§
β’ Since rotations quite small, approximation is valid:
πβ²πβ²πβ²
=πππ
+
π‘π₯π‘π¦π‘π§
MGE-01, Transforming to Official Coordinates, Holst/Medic WS 20/21 7
β’ π π β
β’ π,π, π β πΏ, π, π
πππ
=
(π(π) + β) β cosπ β cos π(π(π) + β) β cosπ β sin π
π2
π2π(π) + β β sinπ
π(π) =π2
π2 β cos2 π + π2 β sin2 π
Hofmann-Wellenhof et al. (2008), Fig.8.1
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β’ π π β
β’ πΏ, π, π β π,π, π
ππβ
=
tanβ1π
π2 + π2β 1 β
π2 β π2
π2β
π(π)
π(π) + β
β1
tanβ1π
π
π2 + π2
cosπβ π(π)
π(π) =π2
π2βπππ 2 π+π2βπ ππ2 π
Hofmann-Wellenhof et al. (2008), Fig.8.1
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β’ π2 =π2βπ2
π2π =
πβπ
π
a 1/f
WGS84 6378137.000 298.25722356300
GRS80 6378137.000 298.25722210088
Bessel 6377387.155 299.15281285000
PZ90.11 6378136.000 298.25783930300
Misra und Enge (2001), Fig. 3.3
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http://libguides.library.arizona.edu/GIS/about-gis
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azimuthal conical cylindrical
www.atlasandboots.com
MGE-01, Transforming to Official Coordinates, Holst/Medic WS 20/21 20
Above: GauΓ-KrΓΌger => tangential
Below: Universal Transversal Mercator (UTM) => intersectional
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Offset to central meridian [km]
Offset to central meridian [km]
Distortion in distance [mm] (distance 100 m, height = 0 m above GRS80 ellipsoid)
0 - 40
100 - 28
180 0
200 + 9
Source: Benning (2009)
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N
E
0 Equator
Fa
cto
r
0,9
99
6
ca. 400 km Latitude
ca. 50Β°
Offset 500 km
East [m] 32 364939
North [m] 5621299
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Fig.: UTM Zones in Europa
Source: www.wikipedia.org
MGE-01, Transforming to Official Coordinates, Holst/Medic WS 20/21 29
a 1/f
WGS84 6378137.000 298.25722356300
GRS80 6378137.000 298.25722210088
Bessel 6377387.155 299.15281285000
PZ90.11 6378136.000 298.25783930300
MGE-01, Transforming to Official Coordinates, Holst/Medic WS 20/21 31
π, π, π
Transition of geodetic datum
Transition of reference surface
WGS84 / ITRF2014
ETRS89
π, π, β
π, π, π π, π, β
πΈ,π π»
Step 3
Step 2
Step 1
Step 4
MGE-01, Transforming to Official Coordinates, Holst/Medic WS 20/21 38
π, π, π
πΏ, π, π
βπ, βπ, βπ
π, π, π
Transition of geodetic datum
WGS84 / ITRF2014
ETRS89 π, π, π
Step 1
βπ, βπ, βπ πΏ + βπΏ, π +βπ, π + βπ
MGE-01, Transforming to Official Coordinates, Holst/Medic WS 20/21 39
β’πππ
=
tanβ1π
π2+π2β 1 β
π2βπ2
π2β
π(π)
π(π)+β
β1
tanβ1π
π
π2+π2
cos πβ π(π)
π(π) =π2
π2βπππ 2 π+π2βπ ππ2 π
πΏ, π, π π, π, π
Transition of geodetic datum
ETRS89 π, π, π π, π, β
Step 2
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β’ π¬ = πΈ0 + 1 β βπΏ + 3 β βπΏ3 + 5 β βπΏ5
β’ π΅ = π β πΊ + 2 β βπΏ2 + 4 β βπΏ4 + 6 β βπΏ6
β’ βπΏ = πΏ β πΏπ πΏπ = 9Β°
β’ πΈ0 =πΏπ+3Β°
6Β°+ 30.5 β 106
β’ π = 0.9996
β’ πΊ = πΊπ β π΅ + πΊ2 β sin 2π΅ + πΊ4 β sin 4π΅ + πΊ6 βsin 6π΅
β’ 1 =π
πβ π β cosπ΅
β’ 3 =π
6π3β π β cos3 π΅ β 1 β π‘2 + π2
β’ 5 =π
120π5β π β cos5 π΅ β
5 β 18π‘2 + π‘4 + π2 β 14 β 58π‘2
β’ 2 =π
2π2β π β cos2 π΅ β π‘
β’ 4 =π
24π4β π β cos4 π΅ β π‘ β 5 β π‘2 + 9 β π2
β’ 6 =π
720π6β π β cos6 π΅ β π‘ β 61 β 58π‘2 + π‘4
β’ π =180Β°
π
β’ π =π
1+π2
β’ π‘ = tanπ΅
β’ π2 =π2βπ2
π2β cos2 π΅
β’ π =π2
π
π,π π¬,π΅
Parameter GRS80 ellipsoid
Go 111β132.952547 m/Β°
G2 -16β038.5088 m
G4 16.8326 m
G6 -0.0220 m
a 6β378β137.000 m
b 6β356β752.314 m
Ξ» = L = Longitude
Ο = B = Latitude
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β
π»
π
β π» = β β π
π π―
Source:
Kavanagh
(2014)
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Source: Federal Office for Cartography and Geodesy, Germany
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A B
X,Y,ZWGS84
X,Y,ZWGS84
X,Y,ZWGS84
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π‘π₯,1, π‘π¦,1,
π‘π§,1, ππ§,1
π‘π₯,2, π‘π¦,2,
π‘π§,2, ππ§,2
π‘π₯,3, π‘π¦,3,
π‘π§,3, ππ§,3
π‘π₯,4, π‘π¦,4,
π‘π§,4, ππ§,4
π‘π, π‘π, π‘π,
ππ,ππ,, ππ
X,Y,ZWGS84 or
X,Y,ZETR89 (absolute positions)
Process each
single laser
scan point
similarly to
GNSS
coordinates;
Start from Step
1 (WGS84) or
Step 2
(ETRS89)
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