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title	summary	tags
The WGS 84 ellipsoid and geodetic coordinates	Latitude, longitude, and height above an ellipsoid, and the closed form conversion to Cartesian coordinates.	geodesy, wgs84, latitude, ellipsoid

The WGS 84 ellipsoid and geodetic coordinates

A position on or near the Earth is usually written as a latitude, a longitude, and a height. To convert that to Cartesian coordinates, the shape of the Earth must be chosen first. The kernels use the WGS 84 ellipsoid.

Definition

The WGS 84 ellipsoid is an oblate ellipsoid of revolution defined by two numbers:

Parameter	Symbol	Value	Unit
Semi-major axis	`a`	6 378 137.0	m
Inverse flattening	`1/f`	298.257 223 563	dimensionless
Flattening	`f`	1 / 298.257 223 563	dimensionless
Semi-minor axis	`b = a (1 - f)`	6 356 752.314 245	m
First eccentricity squared	`e^2 = 2 f - f^2`	0.006 694 379 990 13	dimensionless

Every other quantity is derived from the two defining numbers, so the derived values agree with the published ones to the precision of the definitions rather than being independently rounded.

Three coordinates describe a point:

Geodetic latitude phi, the angle between the ellipsoid normal at the point and the equatorial plane.
Longitude lambda, positive east of the prime meridian.
Height h, measured along the ellipsoid normal, positive outward.

Geodetic latitude is not geocentric latitude. Geocentric latitude is the angle between the radius vector to the point and the equatorial plane. The two agree at the equator and at the poles and differ by up to about 0.19 degrees at mid latitudes, which is about 21 kilometers on the ground.

Mathematical meaning

The conversion from geodetic to Cartesian coordinates is closed form. With


e^2 = 2f - f^2, \qquad
N = \frac{a}{\sqrt{1 - e^2 \sin^2 \phi}}

the Cartesian coordinates in the Earth-fixed frame are


\begin{aligned}
x &= (N + h) \cos\phi \cos\lambda \\
y &= (N + h) \cos\phi \sin\lambda \\
z &= \left( N (1 - e^2) + h \right) \sin\phi
\end{aligned}

N is the distance from the point to the rotation axis measured in the equatorial plane for a point at zero height. It grows from a at the equator to a / sqrt(1 - e^2) at the pole.

The reverse conversion, from Cartesian coordinates to geodetic latitude and height, has no elementary closed form and is solved by iteration or by a series. The implementation uses a method that converges in a fixed number of steps to the precision of the double precision representation. The conversion returns no value when the point lies exactly on the rotation axis, because no longitude is defined there.

The meridian radius of curvature, in the plane of a meridian, is


M = \frac{a (1 - e^2)}{\left(1 - e^2 \sin^2 \phi\right)^{3/2}}

It is the radius of curvature of the ellipsoid in the north-south direction at that latitude. It appears in geodetic computations that integrate along a meridian.

Unit convention

Latitude and longitude are in radians in the interface, and in degrees in the tools, which convert on the boundary.
Height is in meters above the ellipsoid. It may be negative, which places the point below the ellipsoid surface.
The ellipsoid parameters are in meters and are dimensionless where the quantity is a ratio.

Applicability

The WGS 84 ellipsoid is a global average. The actual surface of the Earth differs from it by up to about 100 meters, which is the geoid undulation. A height above the ellipsoid is therefore not a height above mean sea level. The two differ by the geoid undulation at the location, and a caller that needs an orthometric height must apply that correction from a geoid model, which these kernels do not include.

The ellipsoid defines a smooth surface and the conversion is exact for that surface. It does not model the terrain, the atmosphere, or the local deflection of the vertical. The deflection of the vertical, which is the angle between the ellipsoid normal and the true direction of gravity, is below about 0.01 degrees, which is about 1 kilometer over a 10,000 kilometer range, and matters for precise pointing.

Comparison of Earth models

Model	Semi-major axis	Inverse flattening	Intended use	Used here
WGS 84	6 378 137.0 m	298.257 223 563	GPS and modern geodesy	Yes
WGS 72	6 378 135.0 m	298.26	Older satellite work; the SGP4 fit	For the propagator
GRS 80	6 378 137.0 m	298.257 222 101	Geodetic reference	No
Sphere	Any radius	Not applicable	Simple estimates	No

WGS 84 and GRS 80 differ only in the flattening, by about 0.0001 percent, so positions computed with one are indistinguishable from the other at the meter level. WGS 72 differs from WGS 84 in the semi-major axis by 2 meters, which is large enough to matter, and SGP4 was fitted against it.

Source

National Imagery and Mapping Agency, Department of Defense World Geodetic System 1984, Technical Report TR8350.2, third edition, 2000.

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