Geodetic coordinates are a type ofcurvilinearorthogonal coordinate system used ingeodesy based on areference ellipsoid.They includegeodetic latitude (north/south)ϕ,longitude (east/west)λ, andellipsoidal heighth (also known asgeodetic height[1]). The triad is also known asEarth ellipsoidal coordinates[2] (not to be confused withellipsoidal-harmonic coordinates).
Longitude measures the rotationalangle between the zero meridian and the measured point. By convention for the Earth, Moon and Sun, it is expressed in degrees ranging from −180° to +180°. For other bodies a range of 0° to 360° is used.For this purpose, it is necessary to identify azeromeridian, which for Earth is usually thePrime Meridian. For other bodies a fixed surface feature is usually referenced, which for Mars is the meridian passing through the craterAiry-0. It is possible for many different coordinate systems to be defined upon the same reference ellipsoid.
Geodetic latitude measures how close to the poles or equator a point is along a meridian, and is represented as an angle from −90° to +90°, where 0° is the equator. Thegeodetic latitude is the angle between the equatorial plane and a line that isnormal to the reference ellipsoid. Depending on the flattening, it may be slightly different from thegeocentric latitude, which is the angle between the equatorial plane and a line from the center of the ellipsoid. For non-Earth bodies the termsplanetographic latitude andplanetocentric latitude are used instead.
Ellipsoidal height (or ellipsoidalaltitude), also known as geodetic height (or geodetic altitude), is the distance between the point of interest and the ellipsoid surface, evaluated along theellipsoidal normal vector; it is defined as asigned distance such that points inside the ellipsoid have negative height.
Geodetic latitude andgeocentric latitude have different definitions. Geodetic latitude is defined as the angle between theequatorial plane and thesurface normal at a point on the ellipsoid, whereas geocentric latitude is defined as the angle between the equatorial plane and a radial line connecting the centre of the ellipsoid to a point on the surface (see figure). When used without qualification, the term latitude refers to geodetic latitude. For example, the latitude used ingeographic coordinates is geodetic latitude. The standard notation for geodetic latitude isφ. There is no standard notation for geocentric latitude; examples includeθ,ψ,φ′.
Similarly, geodetic altitude is defined as the height above the ellipsoid surface, normal to the ellipsoid; whereasgeocentric altitude is defined as the distance to the reference ellipsoid along a radial line to the geocenter. When used without qualification, as in aviation, the termaltitude refers to geodetic altitude (possibly with further refinements, such as inorthometric heights). Geocentric altitude is typically used inorbital mechanics (seeorbital altitude).
If the impact of Earth'sequatorial bulge is not significant for a given application (e.g.,interplanetary spaceflight), theEarth ellipsoid may be simplified as aspherical Earth, in which case the geocentric and geodetic latitudes are equal and the latitude-dependent geocentric radius simplifies to a global meanEarth's radius (see also:spherical coordinate system).
Given geodetic coordinates, one can compute thegeocentric Cartesian coordinates of the point as follows:[3]
wherea andb are the equatorial radius (semi-major axis) and the polar radius (semi-minor axis), respectively.N is theprime vertical radius of curvature, function of latitudeϕ:
In contrast, extractingϕ,λ andh from the rectangular coordinates usually requiresiteration asϕ andh are mutually involved throughN:[4][5]
where. More sophisticated methods areavailable.