Earth-Fixed Coordinate Systems

From our human perspective, we define location relative to the Earth. Therefore, in a flight simulator, the location of the airplane and every other object is specified relative (directly or indirectly) to a coordinate system fixed to the Earth. The following paragraphs describe various Earth-fixed coordinate systems.

Geocentric Cartesian Coordinates.

Geocentric Cartesian coordinates are fixed to the rotating Earth, originating from the Earth's center. The z-axis points through the geographic North Pole (and coincides with the Earth's axis of rotation). The x-axis points through the intersection of Equator and Prime Meridian ( longitude by latitude). The y-axis completes the right-handed system.

Geocentric Spherical Coordinates.

Because the Earth is spherical, it makes sense to define spherical coordinates for the Earth. The spherical coordinates $(r,\theta,\phi')$ are radius from Earth's center, longitude, and geocentric latitude, respectively. The range of $\theta$ is to , while the range of $\phi$ is to .

The relation between geocentric spherical and geocentric Cartesian coordinates is:

$$x=r\cos\theta\sin\phi' \qquad y=r\sin\theta\sin\phi' \qquad z=r\cos\phi'$$ (1)

Geodetic Coordinates.

Spherical coordinates can be inconvenient for two reasons. First, the geocentric latitude, $\phi'$, is not exactly the same as the geographic latitude used in navigation. This is because the Earth is actually an oblate spheroid, slightly flattened at the poles. And second, radius from the Earth's center is an unwieldy coordinate.

Geodetic coordinates are often more convenient than spherical coordinates. In the geodetic coordinate system, the coordinates $(h,\theta,\phi)$ are altitude, longitude, and latitude. The geodetic latitude and longitude are the same latitude and longitude used in navigation and on maps. The geodetic and geocentric longitudes are the same.

The transformations to and from geodetic coordinates are complex. Reference 1 (pp. 809-810) gives formulas for the transformation from geodetic to geocentric coordinates. The opposite transformation, from geocentric to geodetic coordinates, is very tricky, as it requires the solution of a quartic equation and selection of the proper root. References 2 and 3 give closed-form formulas for this transformation.

Because of the complexity in transforming to and from geodetic coordinates, a flight simulator usually does not use geodetic coordinates internally. Rather, it uses geodetic coordinates only for input and output, and uses a geocentric system internally.

Flat-Earth Coordinates.

In many flight simulators, global navigation is not important. For example, the range of flight could be limited to a small area, or the simulator might not care about the airplane's location.

In such cases, it is appropriate to model the Earth as a plane half-space rather than an oblate spheroid. Then, the simulator need not worry about how the local horizontal plane changes as the airplane flies around the Earth. This simplifies the bookkeeping in the simulator considerably.

The flat-Earth coordinate system is a Cartesian system, which originates at the surface. The z-axis points vertically down, the x-axis points north, and the y-axis points east.