133 lines
5.4 KiB
Markdown
133 lines
5.4 KiB
Markdown
---
|
|
id: geodesy-wgs84
|
|
title: The WGS 84 ellipsoid and geodetic coordinates
|
|
chapter: geometry
|
|
summary: Latitude, longitude, and height above an ellipsoid, and the closed form conversion to Cartesian coordinates.
|
|
tags: 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
|
|
|
|
```text
|
|
e^2 = 2 f - f^2
|
|
N = a / sqrt(1 - e^2 sin^2 phi) (prime vertical radius of curvature)
|
|
```
|
|
|
|
the Cartesian coordinates in the Earth-fixed frame are
|
|
|
|
```text
|
|
x = (N + h) cos(phi) cos(lambda)
|
|
y = (N + h) cos(phi) sin(lambda)
|
|
z = (N (1 - e^2) + h) sin(phi)
|
|
```
|
|
|
|
`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
|
|
|
|
```text
|
|
M = a (1 - e^2) / (1 - e^2 sin^2 phi)^(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.
|
|
|
|
## See also
|
|
|
|
- [[reference-frames]] for the frames the coordinates are expressed in
|
|
- [[geodesy-wgs84]] is used by the propagator only for reporting altitude;
|
|
the propagator itself uses the gravity model in [[sgp4]]
|
|
- Try the conversion in the online tool: *Geodetic position to Earth-fixed
|
|
coordinates*
|