Hi, I just noticed that on this page:
there are a number of references to “sphere” when they should instead say “spheroid”, and to “great circle arc” when they should instead say “geodesic”.
This is clear from the subsequent discussion, in particular in the FAQ section:
where it says
4.2.3.1.

Do you calculate on the sphere or the spheroid?


By default, all distance and area calculations are done on the spheroid. You should find that the results of calculations in local areas match up will with local planar results in good local projections. Over larger areas, the spheroidal
calculations will be more accurate than any calculation done on a projected plane.
All the geography functions have the option of using a sphere calculation, by setting a final boolean parameter to 'FALSE'. This will somewhat speed up calculations, particularly for cases where the geometries are very simple.

So in
4.2. PostGIS Geography Type
the corrected documentation would change:
The basis for the PostGIS geographic type is a sphere. The shortest path between two
points on the sphere is a great circle arc. That means that calculations on geographies (areas, distances, lengths, intersections, etc) must be calculated on the sphere, using more complicated mathematics. For more accurate measurements, the calculations must
take the actual spheroidal shape of the world into account, and the mathematics becomes very complicated indeed.
to:
The basis for the PostGIS geographic type is a spheroid, a “squashed” sphere whose
polar diameter is smaller than its equatorial diameter. The shortest path between two points on the sphere is a great circle arc, and on a spheroid it’s a similar arc called a geodesic. That means that calculations on geographies (areas, distances, lengths,
intersections, etc) will provide more accurate measurements, but the calculations must take the actual spheroidal shape of the world into account, and the mathematics becomes very complicated indeed.
and in
4.2.3.3. What is the longest arc you can process?
the corrected documentation would change:
We use great circle arcs as the "interpolation line" between two points. That means
any two points are actually joined up two ways, depending on which direction you travel along the great circle. All our code assumes that the points are joined by the *shorter* of the two paths along the great circle. As a consequence, shapes that have arcs
of more than 180 degrees will not be correctly modelled.
to:
We use geodesics as the "interpolation line" between two points (these are great
circle arcs on spheres). That means any two points are actually joined up two ways,
depending on which direction you travel between them. All our code assumes that the points are joined by the *shorter* of the two paths. As a consequence, shapes that have arcs of more than 180 degrees will not be correctly modeled.
Cheers,
— Andy