Center and radius of great circle
[lat,lon,radius] = gc2sc(lat0,lon0,az)
[lat,lon,radius] = gc2sc(lat0,lon0,az,angleunits)
mat = gc2sc(...)
[lat,lon,radius] = gc2sc(lat0,lon0,az) converts
a great circle from great circle notation (i.e., lat, lon, azimuth,
where (lat, lon) is on the circle) to small circle notation (i.e.,
lat, lon, radius, where (lat, lon) is the center of the circle and
the radius is 90 degrees, which is a definition of a great circle).
A great circle has two centers and one is chosen arbitrarily. The
other is its antipode. All inputs and outputs are in units of degrees.
[lat,lon,radius] = gc2sc(lat0,lon0,az,angleunits) where
the units of the inputs and outputs, either
mat = gc2sc(...) returns a single output,
mat = [lat lon radius].
Represent a great circle passing through (25ºS,70ºW) on an azimuth of 45º as a small circle:
[lat,lon,radius] = gc2sc(-25,-70,45) lat = -39.8557 lon = 42.9098 radius = 90
A great circle always bisects the sphere. As a demonstration of this statement, consider the Equator, which passes through any point with a latitude of 0º and proceeds on an azimuth of 90º or 270º. Represent the Equator as a small circle:
[lat,lon,radius] = gc2sc(0,-70,270) lat = 90 lon = 0 radius = 90
Not surprisingly, the small circle is centered on the North Pole. As always at the poles, the longitude is arbitrary because of the convergence of the meridians.
Note that the center coordinates returned by this function always lead to one of two possibilities. Since the great circle bisects the sphere, the antipode of the returned point is also a center with a radius of 90º. In the above example, the South Pole would also be a suitable center for the Equator in a small circle.
Great and Small Circles
A small circle is the intersection of a plane with the surface of a sphere. A great circle is a small circle with a radius of 90º.