Editing Spherical circle

{{short description|Mathematical expression of circle like slices of sphere}}
{{Redirect|Small circle|the typographical symbol|Degree symbol}}
{{inline |date=May 2024}}
[[Image:Small circle.svg|right|thumb|Small circle of a sphere.]]
In [[spherical geometry]], a '''spherical circle''' (often shortened to '''circle''') is the [[locus (mathematics)|locus]] of [[point (geometry)|point]]s on a [[sphere]] at constant [[great-circle distance|spherical distance]] (the ''spherical radius'') from a given point on the sphere (the ''pole'' or ''spherical center'').  It is a [[curve]] of constant [[geodesic curvature]] relative to the sphere, analogous to a [[Generalised circle|line or circle]] in the [[Euclidean plane]]; the curves analogous to [[straight lines]] are called ''[[great circle]]s'', and the curves analogous to planar [[circle]]s are called '''small circles''' or '''lesser circles'''. If the sphere is embedded in three-dimensional [[Euclidean space]], its circles are the [[intersection (geometry)|intersections]] of the sphere with [[plane (geometry)|planes]], and the great circles are intersections with planes passing through the [[center (geometry)|center]] of the sphere.

== Fundamental concepts ==

=== Intrinsic characterization ===

A spherical circle with zero geodesic curvature is called a ''great circle'', and is a [[geodesic]] analogous to a straight line in the plane. A great circle separates the sphere into two equal ''[[Hemisphere (geometry)|hemispheres]]'', each with the great circle as its boundary. If a great circle passes through a point on the sphere, it also passes through the [[antipodal point]] (the unique furthest other point on the sphere). For any pair of distinct non-antipodal points, a unique great circle passes through both. Any two points on a great circle separate it into two ''arcs'' analogous to [[line segment]]s in the plane; the shorter is called the ''minor arc'' and is the shortest path between the points, and the longer is called the ''major arc''.

A circle with non-zero geodesic curvature is called a ''small circle'', and is analogous to a circle in the plane. A small circle separates the sphere into two ''spherical disks'' or ''[[spherical cap]]s'', each with the circle as its boundary. For any triple of distinct non-antipodal points a unique small circle passes through all three. Any two points on the small circle separate it into two ''arcs'', analogous to [[circular arc]]s in the plane.

Every circle has two antipodal poles (or centers) intrinsic to the sphere. A great circle is equidistant to its poles, while a small circle is closer to one pole than the other. [[Concentric]] circles are sometimes called ''parallels'', because they each have constant distance to each-other, and in particular to their concentric great circle, and are in that sense analogous to [[parallel line]]s in the plane.

=== Extrinsic characterization ===
[[Image:Esfera-raio-circulomenor.png|right|thumb|<math>BC^2=AB^2+AC^2</math>, where ''C'' is the center of the sphere, ''A'' is the center of the small circle, and ''B'' is a point in the boundary of the small circle. Therefore, knowing the radius of the sphere, and the distance from the plane of the small circle to C, the radius of the small circle can be determined using the Pythagorean theorem.]]

If the sphere is [[Isometry|isometrically]] [[embedding|embedded]] in [[Euclidean space]], the sphere's [[intersection (geometry)|intersection]] with a [[plane (geometry)|plane]] is a circle, which can be interpreted extrinsically to the sphere as a Euclidean circle: a locus of points in the plane at a constant [[Euclidean distance]] (the ''extrinsic radius'') from a point in the plane (the ''extrinsic center''). A great circle lies on a plane passing through the center of the sphere, so its extrinsic radius is equal to the radius of the sphere itself, and its extrinsic center is the sphere's center. A small circle lies on a plane ''not'' passing through the sphere's center, so its extrinsic radius is smaller than that of the sphere and its extrinsic center is an arbitrary point in the interior of the sphere. Parallel planes cut the sphere into parallel (concentric) small circles; the pair of parallel planes tangent to the sphere are tangent at the poles of these circles, and the [[diameter]] through these poles, passing through the sphere's center and perpendicular to the parallel planes, is called the ''axis'' of the parallel circles.

The sphere's intersection with a second sphere is also a circle, and the sphere's intersection with a concentric [[right circular cylinder]] or [[right circular cone]] is a pair of antipodal circles.

== Applications ==
=== Geodesy ===
In the [[geographic coordinate system]] on a globe, the [[parallel (geography)|parallels]] of [[latitude]] are small circles, with the [[Equator]] the only great circle. By contrast, all [[meridian (geography)|meridians]] of [[longitude]], paired with their opposite meridian in the other [[hemisphere of the Earth|hemisphere]], form great circles.

== References ==
* {{cite journal |mode=cs2 |last=
  Allardice |first=Robert Edgar |author-link=Robert Edgar Allardice |year=
  1883 |title=Spherical Geometry |journal=Proceedings of the Edinburgh Mathematical Society |volume=2 |pages=8–16 |doi=10.1017/S0013091500037020 |doi-access=free }} <!-- hathi trust: https://babel.hathitrust.org/cgi/pt?id=inu.30000021035997&view=1up&seq=16 -->
* {{cite book |mode=cs2 |last=
  Casey |first=John |author-link=John Casey (mathematician) |year=
  1889 |title=A treatise on spherical trigonometry |publisher=Hodges, Figgis, & co. |isbn=
 978-1-4181-8047-8 |url=https://archive.org/details/treatiseonspheri00seri/ }}
* {{cite journal |mode=cs2 |last=
  Papadopoulos |first=Athanase |author-link=Athanase Papadopoulos |year=
  2014 |title=On the works of Euler and his followers on spherical geometry |journal=Gaṇita Bhārati |volume=36 |pages=53–108 |arxiv=1409.4736 }}
* {{cite book |mode=cs2 |last1=
  Todhunter |first1=Isaac |authorlink1=Isaac Todhunter |last2=
  Leathem |first2=John Gaston |year=
  1901 |title=Spherical Trigonometry |edition=Revised |publisher=MacMillan |url=https://archive.org/details/sphericaltrigono00todh/ }}

[[Category:Spherical curves]]
[[Category:Circles]]


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