10/8/2009 Prof. M. ElHussainy 1

المساحة ونظم معلومات األراضى شعبة

محمد صفوت محمد الحسيني/ األستاذ الدكتور

9200 سبتمبر

08/10/2009 Prof. Mohamed S. ElHussainy 2

Geodetic & Astronomic Latitudes

Astronomical Latitude also shown

08/10/2009 Prof. Mohamed S. ElHussainy

3

Laplace Stations

( ) φλαα

φλα

sin

sin

GGA −Λ=−

∆=∆

Deviation of the vertical in azimuth α

( )

( )αη−αξ=ζ

αη+αξ−=ψ

cossin

sincos

ξ = Φ - φ

η = (Λ - λ ) cos φ

08/10/2009 Prof. Mohamed S. ElHussainy 4

We have 2 problems:1. Transform GPS positions given in Cartesian X, Y, Z to geodeticφ, λ, h (WGS84) to project onto UTM.

2. Convert GPS geodetic coordinates from WGS84 ellipsoid toHelmert ellipsoid and hence to EGS.

3 dimensional transformations

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1. Cartesian to Geodetic Coordinates

• Recall:3 dimensional Cartesian coordinates (X,Y,Z) to ellipsoidal geodetic coordinates (ϕ,λ, h ).

Bowring’s Forward Transformation

(φ, λ, h) (X,Y,Z)

( )( )( )( )

+−++

=

φυλφυλφυ

sin )1sin cos cos cos

2 hehh

ZYX

08/10/2009 Prof. Mohamed S. ElHussainy 6

2. Change of Ellipsoid• Because GPS coordinates are computed in WGS84 and EGS is on the Helmert ellipsoid

we must be able to transform from one ellipsoid to another.

• This can be done in a number of ways but the easiest is using a 7-parametertransformation of Cartesian coordinates.

Y′X′

Z′Z

X

O

O′

08/10/2009 Prof. Mohamed S. ElHussainy 7

A 7-parameter transformation is essentially a Helmert transformationin 3 dimensions.

Y

We wish to transform one Cartesian coordinate systemto another.

A

08/10/2009 Prof. Mohamed S. ElHussainy 8

ωZ, ωY, ωx are rotation matrices representing rotations aboutthe XG, YG and ZG axes (applied in that order) which make the XG, YG, ZG axes parallel

to the X, Y, Z axes.

P

ZG

YGXG

The vector OP remains fixed in space while we rotate the coordinateaxes by angles ωx, ω y, ω z respectively.

O′ωx

ωZ

ωY

08/10/2009 Prof. Mohamed S. ElHussainy 9

−=

xx

xxx

ωωωωω

cossin0sincos0

001

−=

1000cossin0sincos

zz

zz

z ωωωω

ω

−=

yy

yy

y

ωω

ωωω

cos0sin010

sin0cos

−+−−

+−+=

xyxyy

xyzxzxyzzy

xzxyzzxyzzy

ω cosω cosωsin ω cosωsin ω cosωsin ωsin ω cosω cosωsin ωsin ωsin ωsin ω cos

cosωωsin ω cosωsin ω cosω cosωsin ωsin ωsin ω cosω cosω cosR

X

−−

−=

11

1

xy

xz

yz

Rωω

ωωωω

where ω x, ω y, ωz are the small rotations in radians

08/10/2009 Prof. Mohamed S. ElHussainy 10

7 transformation parameters

where:

(X, Y, Z)A and (X, Y, Z)G are Cartesian coordinates centred on O and O′ respectively

1 + k represents the scale factor to be applied between the twocoordinate systems

ΔX, ΔY, ΔZ is the vector representing the origin shift betweenO′ and O

To transform from one ellipsoid to another we need to knowthe 7 transformation parameters, dX, dY, dZ, k, ω x, ω y, ω z .

Note: for the reverse transformation we need only change thesigns of the seven transformation parameters.

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Transformation Parameters in Egypt

EGD50 to WGS84

• ∆X = - 125.93 m,

• ∆Y = 112.29 m,

• ∆Z = - 9.61 m,• k = 3.14 ppm (for different

scale),

• ωX = 0.17 ” ,

• ωY = 0.13 ” ,

• ωZ = 1.90 ” .• For Translation only:

• ∆X = - 120.31 m, ∆Y = 123.77 m, ∆Z = - 8.67 m.

EGD80 to WGS84

• ∆X = - 75.03 m,

• ∆Y = 31.42 m,

• ∆Z = - 15.54 m,• k = 3.14 ppm (for different scale),

• ωX = 0.17 sec,

• ωY = 0.13 sec,

• ωZ = 1.90 sec (∆ A = 0, ∆ξ = -3.6, ∆ η = - 1.1 sec) .

• For Translation only:

• ∆X = - 70.69 m, ∆Y = 33.63 m, ∆Z = - 14.42 m.

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Example:

convert WGS84 coordinates:

=

380.3075797391.2967106686.4728037

ZYX

to EGD

1. Form R matrix (convert angles to radians)

−−−−−−−−−

=1697.06485.0

697.0167.96485.067.91

EEEEEE

R

ωX = - 0.2, ωY = - 0.1, ωZ = - 2 sec

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2. Apply matrix and origin shift

−−−−−−−−−−−−−−−

+

−=

380.3075797391.2967106686.4728037

.631697.06485.0697.063167.96485.067.9631

1531

75

84EEEEEEEEE

ZYX

WGS

X = 4 728 070.56Y = 2 967 108.94Z = 3 075 803.31 EGD

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WGS84 Cartesian to EGS

7 parameter

XYZWGS84

E NEGS

φ λ hEGD

XYZEGS

Bowring

Redfearn

Note: height transformed by hWGS84 -N = HEHD