Editing Great circle

{{short description|Spherical geometry analog of a straight line}}
{{Redirect|Great Circle}}{{More citations needed|date=April 2025}}[[File:Great circle, axis, and poles.svg|thumb|upright=1.25|The great circle {{mvar|g}} (green) lies in a plane through the sphere's center {{mvar|O}} (black). The perpendicular line {{mvar|a}} (purple) through the center is called the ''axis'' of {{mvar|g}}, and its two intersections with the sphere, {{mvar|P}} and {{math|''P''{{'}}}} (red), are the ''poles'' of {{mvar|g}}. Any great circle {{mvar|s}} (blue) through the poles is ''secondary'' to {{mvar|g}}.]]
[[File:Great circle hemispheres.png|thumb|right|A great circle divides the sphere in two equal hemispheres.]]

In [[mathematics]], a '''great circle''' or '''orthodrome''' is the [[circle|circular]] [[Intersection (geometry)|intersection]] of a [[sphere]] and a [[Plane (geometry)|plane]] [[incidence (geometry)|passing through]] the sphere's [[centre (geometry)|center point]].<ref>{{Cite web |last=W. |first=Weisstein, Eric |title=Great Circle -- from Wolfram MathWorld |url=https://mathworld.wolfram.com/GreatCircle.html |access-date=2022-09-30 |website=mathworld.wolfram.com |language=en}}</ref><ref>{{Cite book |last1=Weintrit |first1=Adam |url=https://dl.acm.org/doi/abs/10.5555/2788309 |title=Loxodrome (Rhumb Line), Orthodrome (Great Circle), Great Ellipse and Geodetic Line (Geodesic) in Navigation |last2=Kopcz |first2=Piotr |date=2014 |publisher=CRC Press, Inc. |isbn=978-1-138-00004-9 |location=USA }}</ref>

==Discussion==
Any [[Circular arc|arc]] of a great circle is a [[geodesic]] of the sphere, so that great circles in [[spherical geometry]] are the natural analog of [[Line (geometry)|straight lines]] in [[Euclidean space]]. For any pair of distinct non-[[Antipodal point|antipodal]] [[point (geometry)|point]]s on the sphere, there is a unique great circle passing through both. (Every great circle through any point also passes through its antipodal point, so there are infinitely many great circles through two antipodal points.) The shorter of the two great-circle arcs between two distinct points on the sphere is called the ''minor arc'', and is the shortest surface-path between them. Its [[arc length]] is the [[great-circle distance]] between the points (the [[intrinsic metric|intrinsic distance]] on a sphere), and is proportional to the [[angle measure|measure]] of the [[central angle]] formed by the two points and the center of the sphere.

A great circle is the largest circle that can be drawn on any given sphere. Any [[diameter]] of any great circle coincides with a diameter of the sphere, and therefore every great circle is [[Concentric objects|concentric]] with the sphere and shares the same [[radius]]. Any other [[circle of a sphere|circle of the sphere]] is called a [[small circle]], and is the intersection of the sphere with a plane not passing through its center. Small circles are the spherical-geometry analog of circles in Euclidean space.

Every circle in Euclidean 3-space is a great circle of exactly one sphere.

The [[disk (mathematics)|disk]] bounded by a great circle is called a ''great disk'': it is the intersection of a [[ball (geometry)|ball]] and a plane passing through its center.
In higher dimensions, the great circles on the [[n-sphere|''n''-sphere]] are the intersection of the ''n''-sphere with 2-planes that pass through the origin in the [[Euclidean space]] {{math|'''R'''<sup>''n'' + 1</sup>}}.

Half of a great circle may be called a ''great [[semicircle]]'' (e.g., as in parts of a [[Meridian (astronomy)|meridian in astronomy]]).

==Derivation of shortest paths==
{{see also|Great-circle distance}}
To prove that the minor arc of a great circle is the shortest path connecting two points on the surface of a sphere, one can apply [[calculus of variations]] to it.

Consider the class of all regular paths from a point <math>p</math> to another point <math>q</math>. Introduce [[spherical coordinates]] so that <math>p</math> coincides with the north pole. Any curve on the sphere that does not intersect either pole, except possibly at the endpoints, can be parametrized by

:<math>\theta = \theta(t),\quad \phi = \phi(t),\quad a\le t\le b</math>

provided <math>\phi</math> is allowed to take on arbitrary real values. The infinitesimal arc length in these coordinates is

: <math>
ds=r\sqrt{\theta'^2+\phi'^{2}\sin^{2}\theta}\, dt.
</math>

So the length of a curve <math>\gamma</math> from <math>p</math> to <math>q</math> is a [[functional (mathematics)|functional]] of the curve given by

: <math>
S[\gamma]=r\int_a^b\sqrt{\theta'^2+\phi'^{2}\sin^{2}\theta}\, dt.
</math>

According to the [[Euler–Lagrange equation]], <math>S[\gamma]</math> is minimized if and only if
:<math> \frac{\sin^2\theta\phi'}{\sqrt{\theta'^2+\phi'^2\sin^2\theta}}=C</math>,
where <math>C</math> is a <math>t</math>-independent constant, and
:<math> \frac{\sin\theta\cos\theta\phi'^2}{\sqrt{\theta'^2+\phi'^2\sin^2\theta}}=\frac{d}{dt}\frac{\theta'}{\sqrt{\theta'^2+\phi'^2\sin^2\theta}}.</math>
From the first equation of these two, it can be obtained that
:<math> \phi'=\frac{C\theta'}{\sin\theta\sqrt{\sin^2\theta-C^2}}</math>.
Integrating both sides and considering the boundary condition, the real solution of <math>C</math> is zero. Thus, <math>\phi'=0</math> and <math>\theta</math> can be any value between 0 and <math>\theta_0</math>, indicating that the curve must lie on a meridian of the sphere. In a [[Cartesian coordinate system]], this is
:<math>x\sin\phi_0 - y\cos\phi_0 = 0</math>
which is a plane through the origin, i.e., the center of the sphere.

==Applications==
Some examples of great circles on the [[celestial sphere]] include the [[celestial horizon]], the [[celestial equator]], and the [[ecliptic]]. Great circles are also used as rather accurate approximations of [[geodesics on an ellipsoid|geodesics]] on the [[Earth]]'s surface for air or sea [[Great-circle navigation|navigation]] (although it [[shape of the Earth|is not a perfect sphere]]), as well as on spheroidal [[celestial bodies]].

The [[equator]] of the idealized earth is a great circle and any meridian and its opposite meridian form a great circle. Another great circle is the one that divides the [[land and water hemispheres]]. A great circle divides the earth into two [[hemisphere of the Earth|hemispheres]] and if a great circle passes through a point it must pass through its [[antipodal point]].

The [[Funk transform]] integrates a function along all great circles of the sphere.

==See also==
* [[Great ellipse]]
* [[Rhumb line]]

== References ==
{{Reflist}}

==External links==
* [http://mathworld.wolfram.com/GreatCircle.html Great Circle – from MathWorld] Great Circle description, figures, and equations. Mathworld, Wolfram Research, Inc. c1999
* [http://demonstrations.wolfram.com/GreatCirclesOnMercatorsChart/ Great Circles on Mercator's Chart] by John Snyder with additional contributions by Jeff Bryant, Pratik Desai, and Carl Woll, [[Wolfram Demonstrations Project]].

[[Category:Elementary geometry]]
[[Category:Spherical trigonometry]]
[[Category:Riemannian geometry]]
[[Category:Circles]]
[[Category:Spherical curves]]