On a sphere or ellipsoid, the central angle is delineated along a great circle. The usually provided coordinates of a point on a sphere/ellipsoid is its conjugate latitude ("Lat"), , and longitude ("Long"), . The "point", , is actually—relative to the great circle it is being measured on—the transverse colatitude ("TvL"), and the central angle/angular distance is the difference between two TvLs, .
Calculation of TvL
The calculation of and can be found using a common subroutine:
Each point has at least two values, both a forward and reverse value.
Occupying great circle
The arc path, , tracing the great circle that a central angle occupies, is measured as that great circle's azimuth at the equator, introducing an important property of spherical geometry, Clairaut's constant:
From this and relationships to ,
Angular distance formulary
The angular distance can be calculated either directly as the TvL difference, or via the common coordinates (here, either SAw, SBw value set can be used):
and, using half-angles,
Also, by using Cartesian products rather than differences, the origin of the spherical cosine for sides becomes apparent:
There is also a logarithmical form: