# EPSG:9837

## Geographic/topocentric conversions

### Attributes

Data source: OGP

Information source: EPSG guidance note #7-2, http://www.epsg.org

Revision date: 2018-08-29

#### Export

Definition: OGP XML

```<?xml version="1.0" encoding="UTF-8"?>
<epsg:informationSource>EPSG guidance note #7-2, http://www.epsg.org</epsg:informationSource>
<epsg:revisionDate>2018-08-29</epsg:revisionDate>
<epsg:changes>
</epsg:changes>
<epsg:show>true</epsg:show>
<epsg:isDeprecated>false</epsg:isDeprecated>
<epsg:isOperationReversible>true</epsg:isOperationReversible>
<epsg:example>For Geographic 3D CRS = WGS 84 (EPSG CRS code 4979)
and
Topocentric origin latitude Po =  55deg N
Topocentric origin longitude Lo = 5 deg E
Topocentric origin ellipsoidal height ho = 	200 metres

Ellipsoid parameters: 	= 6378137.0 metres, 1/f = 298.25722236

First calculate additional ellipsoid parameter e^2 and radius of curvature nuO at the topocentric origin:
e^2=0.006694380
nuO	=	0. 6392510.727

Forward calculation for:
Latitude P = 	53°48'382"N
Longitude L = 	2°07'468"E
Height h 	=	73.0 metres

nu	=	6392088.017
then
U 	 =	–189 013.869 m
V 	 =	–128 642.040 m
W 	=	   – 4 220.171 m

Reverse calculation for:
U 	=	–189 013.869 m
V 	=	–128 642.040 m
W 	=	   – 4 220.171 m

First calculate additional ellipsoid parameter e^2 and radius of curvature nuO at the topocentric origin:
e^2 = 0.006694380
nuO = 6392510.727

then the following intermediate terms:

Xo = 3652755.306
Yo = 319574.680
Zo = 5201547.353
X = 3771793.968
Y = 140253.342
Z = 5124304.349
eta = 0.006739496674
b = 6356752.314
p = 3774400.712
q = .937549875
nu = 6392088.017

for a final result of:

Latitude P = 	53°48'33.820"N
Longitude L 	=	2°07'46.380"E
Height h = 	73.0 metres</epsg:example>
<gml:identifier codeSpace="IOGP">urn:ogc:def:method:EPSG::9837</gml:identifier>
<gml:name>Geographic/topocentric conversions</gml:name>
<gml: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.

To convert latitude P, longitude L and ellipsoidal height h into topocentric coordinates U,V,W:

U = (nu + h) cos(P) sin (L – Lo)
V = (nu + h) [sin(P) cos(Po) – cos(P) sin(Po) cos(L – Lo)] + e^2 (nuO sin(Po) – nu sin(P)) cos(Po)
W = (nu + h) [sin(P) sin(Po) + cos(P) cos(Po) cos(L – Lo)] + e^2 (nuO sin(Po) – nu sin(P)) sin(Po) – (nuO – ho)

where Po, Lo and ho are the ellipsoidal coordinates of the topocentric origin

and nu is the radius of curvature in the prime vertical at latitude P = a /(1 – e^2 sin^2 P)^0.5
nuO is the radius of curvature in the prime vertical at latitude Po = a /(1 – e^2 sin^2 Po)^0.5
e is the eccentricity of the ellipsoid here e^2 = (a^2 – b^2)/a^2 = 2f – f^2

The reverse formulae to convert topocentric coordinates (U, V, W) into latitude, longitude and ellipsoidal height (P, L, h) first draws on the reverse case of method 9836 to derive geocentric coordinates X, Y, Z and then on the reverse case of method 9602 to derive latitude, longitude and height.

First,
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)

where,
XO = (nuO + hO) cos(Po) cos(Lo)
YO = (nuO + hO) cos(Po) sin(Lo)
ZO = [(1 – e2) nuO + hO] sin(Po)
Po, Lo, hO are the ellipsoidal coordinates of the topocentric origin,
nuO is the radius of curvature in the prime vertical at latitude Po = a /(1 – e^2 sin^2(Po))^0.5, and
e is the eccentricity of the ellipsoid where e^2 = (a^2 – b^2)/a^2 = 2f – f^2.

Then,
lat = atan2[(Z + ε b sin^3(q)) , (p – e^2 a cos^3(q))]
lon = atan2 (Y , X)
where
ε = e^2 / (1 – e^2)
b = a(1 – f)
p = (X^2 + Y^2)^0.5
q = atan2[(Z a) , (p b)]
L is relative to the Greenwich prime meridian.

and
h = (p / cos(P)) – nu
where
nu is the radius of curvature in the prime vertical at latitude lat = a /(1 – e^2sin^2(P))^0.5
(see implementation notes in preface for atan2 convention)</gml:formula>