Geocentric/topocentric conversions

Formula

Note: These formulas have been transcribed from EPSG Guidance Note #7-2. Users are encouraged to use that document rather than the text which follows as reference because limitations in the transcription will be avoided.

First it is necessary to derive ellipsoidal values Po, Lo of the topocentric origin from their geocentric values Xo, Yo, Zo through the reverse formulas given in method 9602.

Then topocentric coordinates [U, V, W] are computed as follows:

U = – (X-Xo) sin Lo + (Y-Yo) cos Lo
V = – (X–Xo) sin Po cos Lo – (Y–Yo) sin Po sin Lo + (Z–Zo) cos Po
W = (X–Xo) cos Po cos Lo + (Y–Yo) cos Po sin Lo + (Z–Zo) sin Po 

The reverse formulas to calculate geocentric coordinates from topocentric coordinates are:

X = Xo – U sin Lo  – V sin Po cos Lo + W cos Po cos Lo
Y = Yo + U cos Lo – V sin Po sin Lo  + W cos Po sin Lo 
Z = Zo + V cos Po + W sin Po

Example

For Geocentric CRS = 	WGS 84 (EPSG CRS code 4978)
and		
Topocentric origin Xo 	=	3652 755.3058 m
Topocentric origin Yo 	=	  319 574.6799 m
Topocentric origin Zo	=	5201 547.3536 m

Ellipsoid parameters:	a = 6378137m.0  1/f = 298.2572236

First calculate additional ellipsoid parameters:
e^2 = 0.006694380
eta = 0.006739497
b = 6356752.314

Next, derive Po, Lo from Xo,Yo,Zo by the formulas given in method 9602:
p	=	3666708.2376	
q	=	0.9583523313	
Po	=	0.9599310885	rad
Lo	=	0.0872664625	rad

Forward calculation for point with geocentric coordinates:
X= 3771 793.968
Y=  140 253.342
Z= 5124 304.349

gives topocentric coordinates
U= -189 013.869
V= -128 642.040
W=    -4 220.171

EPSG:9836

Geocentric/topocentric conversions

Attributes

Covered area powered by MapTiler

Formula

Example