Point motion (ellipsoidal)

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.

The ellipsoidal coordinates of a point at coordinate epoch t1 may be calculated at any other coordinate epoch t2 from:

[Lat(t2)] = [Lat(t1)] + (t2-t1) * [V_lat]
[Lon(t2)] = [Lon(t1)] + (t2-t1) * [V_lon]
[h(t2)] = [h(t1)] + (t2-t1) * [V_up]

where V describes the velocities of the ellipsoidal coordinates. These are usually given as a rate in linear units (say millimetres per year) resolved into north, east and up components VN, VE and Vh. The north and east components are converted into latitude and longitude components by:
	V_lat = VN / (rho+h)
	V_lon = VE / [(nu+h) cos(lat)] 
where 
	rho is radius of curvature of the CRS's ellipsoid in the plane of the meridian at latitude lat, where 
	rho = a(1 – e^2)/(1 – e^2sin^2(lat))^3/2
	nu is radius of curvature of the ellipsoid perpendicular to the meridian at latitude lat, where 
	nu = a /(1 – e^2sin^2(lat))^1/2

Example

Given a point with ellipsoidal coordinates at coordinate epoch 2017.55 and having linear velocities: 
lat = 51.0°N		VN  = +15.12 mm/yr
lon = 141.0°W		VE  =  –2.86 mm/yr
h = 1000 m		V_up =  +1.10 mm/yr
t = 2017.55 (years)				
whose coordinates are required at epoch 1997.00. The point is referenced to a CRS which uses the GRS 1980 ellipsoid for which a = 6378137.0m and e^2 = 0.006694380.

Then (after conversion to appropriate units):
	V_lat = 2.37174E-09 rad/yr
	V_lon = -7.10972E-07 rad/yr
	V_up = 0.0011 m / yr
and
	lat = 0.890117919 + (1997.00-2017.55) * 2.37174E-09 = 0.890117870 radians = 50°59'59.990"N
	lon = 0.890117919 + (1997.00-2017.55) * -7.10972E-07 = -2.460914231 radians = 140°59'59.997"W
	h = 1000.0 + (1997.00-2017.55) * 0.0011 = 999.977m
	t = 1997.00

EPSG:1067

Point motion (ellipsoidal)

Attributes

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Formula

Example