Editing Spherical geometry (section)

==Principles==
In [[Euclidean geometry|plane (Euclidean) geometry]], the basic concepts are [[Point (geometry)|point]]s and (straight) [[line (mathematics)|line]]s. In spherical geometry, the basic concepts are point and [[great circle]]. However, two great circles on a plane intersect in two antipodal points, unlike coplanar lines in [[Elliptic geometry]].

In the extrinsic 3-dimensional picture, a great circle is the intersection of the sphere with any plane through the center.  In the intrinsic approach, a great circle is a [[geodesic]]; a shortest path between any two of its points provided they are close enough. Or, in the (also intrinsic) axiomatic approach analogous to Euclid's axioms of plane geometry, "great circle" is simply an undefined term, together with postulates stipulating the basic relationships between great circles and the also-undefined "points".  This is the same as Euclid's method of treating point and line as undefined primitive notions and axiomatizing their relationships.  

Great circles in many ways play the same logical role in spherical geometry as lines in Euclidean geometry, e.g., as the sides of (spherical) triangles.  This is more than an analogy; spherical and plane geometry and others can all be unified under the umbrella of geometry [[Riemannian geometry|built from distance measurement]], where "lines" are defined to mean shortest paths (geodesics). Many statements about the geometry of points and such "lines" are equally true in all those geometries provided lines are defined that way, and the theory can be readily extended to higher dimensions.   Nevertheless, because its applications and pedagogy are tied to solid geometry, and because the generalization loses some important properties of lines in the plane, spherical geometry ordinarily does not use the term "line" at all to refer to anything on the sphere itself. If developed as a part of solid geometry, use is made of points, straight lines and planes (in the Euclidean sense) in the surrounding space.

In spherical geometry, [[angle]]s are defined between great circles, resulting in a [[spherical trigonometry]] that differs from ordinary [[trigonometry]] in many respects; for example, the sum of the interior angles of a spherical [[triangle]] exceeds 180 degrees.