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