Editing Sphere (section)

==Treatment by area of mathematics==
===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]].

===Differential geometry===
The sphere is a [[smooth surface]] with constant [[Gaussian curvature]] at each point equal to {{math|1/''r''<sup>2</sup>}}.<ref name=MathWorld_Sphere /> As per Gauss's [[Theorema Egregium]], this curvature is independent of the sphere's embedding in 3-dimensional space. Also following from Gauss, a sphere cannot be mapped to a plane while maintaining both areas and angles. Therefore, any [[map projection]] introduces some form of distortion.

A sphere of radius {{mvar|r}} has [[area element]] <math>dA = r^2 \sin \theta\, d\theta\, d\varphi</math>. This can be found from the [[volume element]] in [[spherical coordinates]] with {{mvar|r}} held constant.<ref name=MathWorld_Sphere />

A sphere of any radius centered at zero is an [[integral surface]] of the following [[differential form]]:
:<math> x \, dx + y \, dy + z \, dz = 0.</math>
This equation reflects that the position vector and [[tangent plane (geometry)|tangent plane]] at a point are always [[Orthogonality|orthogonal]] to each other. Furthermore, the outward-facing [[normal vector]] is equal to the position vector scaled by {{mvar|1/r}}.

In [[Riemannian geometry]], the [[filling area conjecture]] states that the hemisphere is the optimal (least area) isometric filling of the [[Riemannian circle]].

===Topology===
Remarkably, it is possible to turn an ordinary sphere inside out in a [[three-dimensional space]] with possible self-intersections but without creating any creases, in a process called [[sphere eversion]].

The antipodal quotient of the sphere is the surface called the [[real projective plane]], which can also be thought of as the [[Northern Hemisphere]] with antipodal points of the equator identified.