(03)Transformation WGS84 to PRS92 and Vice Versa

Solution Solve for the first eccentricity using WGS84 parameters f = exe = 2f - (fxf) exe =

1/

0.00669438

Solve for the radius of the prime vertical ellipse, N a = φ = N = a/ sqrt (1-((e x e) x (sin φ) x (sin φ)) N = 6378137/ N = Compute the Cartesian coordinates, X, Y and Z (in meters): h = λ = X = (N+h) cosφ cosλ Y = (N+h) cosφ sinλ Z = ((N(1-(e x e)))+h) sinφ X = Y = Z = Compute for the scale in parts per million Scale to WGS84 = Scale from WGS84 = -6 (1+Sc*10 ) =

298.257223563

6378137 13d 33' 35.89825"

13.55997174

0.999815978 6379310.93163

319.7403 121d 52' 7.97792"

121.8688828

-3274408.20902 5266930.62734 1485774.60292

-1.06002 1.06002

1.353352832 1.00000106002

Perform the matrix multiplication of the rotation matrix and the cartesian coordinates in arc seconds in radians Rotation to WGS84 Rx = 3.06762 1.4872241E-05 Ry = -4.90291 -2.3769978E-05

Rz

=

-1.5779

-7.6498751E-06

= = =

-3.06762 4.90291 1.5779

-1.4872241E-05 2.3769978E-05 7.6498751E-06

1 -7.6498751E-06 2.3769978E-05

7.6498751E-06 1 1.4872241E-05

-2.3769978E-05 -1.4872241E-05 1

X

-3274403.234492 5266933.579354 1485775.101369

-3274403.234 5266933.579 1485775.101

Rotation from WGS84 to PRS92 Rx Ry Rz

R

Perform the matrix multiplication of the scale matrix and the output from the previous result -3274406.70542480 5266939.16240858 1485776.67631992 Perform the matrix addition Translation to WGS84 ∆X ∆Y ∆Z

= = =

-127.62195 -67.24478 -47.04305

=

-3274408.20902 5266930.62734 1485774.60292

Translation from WGS84 to PRS92 ∆X ∆Y ∆Z

= = =

127.62195 67.24478 47.04305 -3274279.08347480 =X2 5267006.40718858 =Y2 1485823.71937 =Z2

Solve for the Longitude λ

Y   X

λ = tan −1 

λ

=

dd

tan-1

mm

λ=

121

-1.608600328 -1.014603527 rad -58.13249999 degrees 121.86750001 degrees ss

52

3.00000000

Solve for D

D =

X

2

+Y2

D

6201794.90227194

Get an initial estimate of the latitude, φ 0 first eccentricity using PRS92 parameters f exe exe

φ = tan −1

Z

= = 2f - (fxf) =

1/

294.980000000 0.0067686281770

φ0 = tan −1

Z D (1 − e 2 )

φ0

=

tan-1

0.24121230501 0.236690944 rad 13.56139216 degrees

φ0 dd φ0 =

mm

ss

13

33

41.01176000

Estimate the value of the prime vertical ellipse, N 0 and f 0 (Iteration) Use value of a from PRS92 a Iteration 1

N0 =

=

6378206.4

a 1 − e sin 2 φ0 2

N0

φ0 = tan

−1

=

6379393.61249109

Z + e 2 N 0 sin φ0 D φ0

=

φ0

= dd

tan-1

mm

φ1 = Iteration 2 N1

13

φ2

=

φ2

=

0.241212236 0.236690879 rad 13.5613884 degrees ss

33

40.99823000

=

dd

6379393.61185 tan-1

mm

0.241212235 0.236690878 rad 13.56138837 degrees ss

φ2 = Iteration 3 N2

13

φ3

=

40.99814000

=

φ3

6379393.61184 tan-1

0.241212235 0.236690878 rad 13.56138837 degrees

= dd

mm

φ3 = Iteration 4 N3

13

φ4

=

ss 33

40.99814000

=

φ4

6379393.61184 tan-1

0.241212235 0.236690878 rad 13.56138837 degrees

= dd

φ4=

33

mm 13

ss 33

40.99814000

Compute h

h=

D − N0 cos φ0 h

=

271.1797203