Editing Sphere (section)

===Spherical geometry===
{{Main|Spherical geometry}}
[[File:Sphere halve.png|thumb|[[Great circle]] on a sphere]]

The basic elements of [[Euclidean plane geometry]] are [[Point (geometry)|points]] and [[line (mathematics)|lines]]. On the sphere, points are defined in the usual sense. The analogue of the "line" is the [[geodesic]], which is a [[great circle]]; the defining characteristic of a great circle is that the plane containing all its points also passes through the center of the sphere. Measuring by [[arc length]] shows that the shortest path between two points lying on the sphere is the shorter segment of the [[great circle]] that includes the points.

Many theorems from [[classical geometry]] hold true for spherical geometry as well, but not all do because the sphere fails to satisfy some of classical geometry's [[postulate]]s, including the [[parallel postulate]]. In [[spherical trigonometry]], [[angle]]s are defined between great circles. Spherical trigonometry differs from ordinary [[trigonometry]] in many respects. For example, the sum of the interior angles of a [[spherical triangle]] always exceeds 180&nbsp;degrees. Also, any two [[similar triangles|similar]] spherical triangles are congruent.

Any pair of points on a sphere that lie on a straight line through the sphere's center (i.e., the diameter) are called [[antipodal point|''antipodal points'']]{{snd}}on the sphere, the distance between them is exactly half the length of the circumference.{{NoteTag |group="Notes" |It does not matter which direction is chosen, the distance is the sphere's radius × ''π''.}} Any other (i.e., not antipodal) pair of distinct points on a sphere
*lie on a unique great circle,
*segment it into one minor (i.e., shorter) and one major (i.e., longer) [[Arc (geometry)|arc]], and
*have the minor arc's length be the ''shortest distance'' between them on the sphere.{{NoteTag |group="Notes" |The distance between two non-distinct points (i.e., a point and itself) on the sphere is zero.}}

Spherical geometry is a form of [[elliptic geometry]], which together with [[hyperbolic geometry]] makes up [[non-Euclidean geometry]].