133 lines
5.4 KiB
Markdown
133 lines
5.4 KiB
Markdown
---
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title: The WGS 84 ellipsoid and geodetic coordinates
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summary: Latitude, longitude, and height above an ellipsoid, and the closed form conversion to Cartesian coordinates.
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tags: geodesy, wgs84, latitude, ellipsoid
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---
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# The WGS 84 ellipsoid and geodetic coordinates
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A position on or near the Earth is usually written as a latitude, a
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longitude, and a height. To convert that to Cartesian coordinates, the shape
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of the Earth must be chosen first. The kernels use the WGS 84 ellipsoid.
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## Definition
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The WGS 84 ellipsoid is an oblate ellipsoid of revolution defined by two
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numbers:
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| Parameter | Symbol | Value | Unit |
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|-----------|--------|-------|------|
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| Semi-major axis | `a` | 6 378 137.0 | m |
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| Inverse flattening | `1/f` | 298.257 223 563 | dimensionless |
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| Flattening | `f` | 1 / 298.257 223 563 | dimensionless |
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| Semi-minor axis | `b = a (1 - f)` | 6 356 752.314 245 | m |
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| First eccentricity squared | `e^2 = 2 f - f^2` | 0.006 694 379 990 13 | dimensionless |
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Every other quantity is derived from the two defining numbers, so the
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derived values agree with the published ones to the precision of the
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definitions rather than being independently rounded.
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Three coordinates describe a point:
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- **Geodetic latitude** `phi`, the angle between the ellipsoid normal at the
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point and the equatorial plane.
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- **Longitude** `lambda`, positive east of the prime meridian.
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- **Height** `h`, measured along the ellipsoid normal, positive outward.
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Geodetic latitude is not geocentric latitude. Geocentric latitude is the
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angle between the radius vector to the point and the equatorial plane. The
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two agree at the equator and at the poles and differ by up to about 0.19
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degrees at mid latitudes, which is about 21 kilometers on the ground.
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## Mathematical meaning
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The conversion from geodetic to Cartesian coordinates is closed form. With
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$$
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e^2 = 2f - f^2, \qquad
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N = \frac{a}{\sqrt{1 - e^2 \sin^2 \phi}}
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$$
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the Cartesian coordinates in the Earth-fixed frame are
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$$
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\begin{aligned}
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x &= (N + h) \cos\phi \cos\lambda \\
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y &= (N + h) \cos\phi \sin\lambda \\
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z &= \left( N (1 - e^2) + h \right) \sin\phi
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\end{aligned}
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$$
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`N` is the distance from the point to the rotation axis measured in the
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equatorial plane for a point at zero height. It grows from `a` at the
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equator to `a / sqrt(1 - e^2)` at the pole.
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The reverse conversion, from Cartesian coordinates to geodetic latitude and
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height, has no elementary closed form and is solved by iteration or by a
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series. The implementation uses a method that converges in a fixed number of
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steps to the precision of the double precision representation. The
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conversion returns no value when the point lies exactly on the rotation
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axis, because no longitude is defined there.
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The meridian radius of curvature, in the plane of a meridian, is
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$$
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M = \frac{a (1 - e^2)}{\left(1 - e^2 \sin^2 \phi\right)^{3/2}}
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$$
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It is the radius of curvature of the ellipsoid in the north-south direction
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at that latitude. It appears in geodetic computations that integrate along a
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meridian.
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## Unit convention
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- Latitude and longitude are in **radians** in the interface, and in
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**degrees** in the tools, which convert on the boundary.
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- Height is in **meters** above the ellipsoid. It may be negative, which
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places the point below the ellipsoid surface.
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- The ellipsoid parameters are in **meters** and are dimensionless where the
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quantity is a ratio.
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## Applicability
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The WGS 84 ellipsoid is a global average. The actual surface of the Earth
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differs from it by up to about 100 meters, which is the geoid undulation. A
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height above the ellipsoid is therefore not a height above mean sea level.
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The two differ by the geoid undulation at the location, and a caller that
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needs an orthometric height must apply that correction from a geoid model,
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which these kernels do not include.
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The ellipsoid defines a smooth surface and the conversion is exact for that
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surface. It does not model the terrain, the atmosphere, or the local
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deflection of the vertical. The deflection of the vertical, which is the
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angle between the ellipsoid normal and the true direction of gravity, is
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below about 0.01 degrees, which is about 1 kilometer over a 10,000
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kilometer range, and matters for precise pointing.
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## Comparison of Earth models
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| Model | Semi-major axis | Inverse flattening | Intended use | Used here |
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|-------|-----------------|--------------------|--------------|-----------|
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| WGS 84 | 6 378 137.0 m | 298.257 223 563 | GPS and modern geodesy | Yes |
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| WGS 72 | 6 378 135.0 m | 298.26 | Older satellite work; the SGP4 fit | For the propagator |
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| GRS 80 | 6 378 137.0 m | 298.257 222 101 | Geodetic reference | No |
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| Sphere | Any radius | Not applicable | Simple estimates | No |
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WGS 84 and GRS 80 differ only in the flattening, by about 0.0001 percent, so
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positions computed with one are indistinguishable from the other at the
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meter level. WGS 72 differs from WGS 84 in the semi-major axis by 2 meters,
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which is large enough to matter, and SGP4 was fitted against it.
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## Source
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- National Imagery and Mapping Agency, *Department of Defense World Geodetic
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System 1984*, Technical Report TR8350.2, third edition, 2000.
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## See also
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- [[reference-frames]] for the frames the coordinates are expressed in
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- [[geodesy-wgs84]] is used by the propagator only for reporting altitude;
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the propagator itself uses the gravity model in [[sgp4]]
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- Try the conversion in the online tool: *Geodetic position to Earth-fixed
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coordinates*
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