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Great circle

From Wikipedia, the free encyclopedia
Spherical geometry analog of a straight line
"Great Circle" redirects here. For other uses, seeGreat Circle (disambiguation).
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The great circleg (green) lies in a plane through the sphere's centerO (black). The perpendicular linea (purple) through the center is called theaxis ofg, and its two intersections with the sphere,P andP' (red), are thepoles ofg. Any great circles (blue) through the poles issecondary tog.
A great circle divides the sphere in two equal hemispheres.

Inmathematics, agreat circle ororthodrome is thecircularintersection of asphere and aplanepassing through the sphere'scenter point.[1][2]

Discussion

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Anyarc of a great circle is ageodesic of the sphere, so that great circles inspherical geometry are the natural analog ofstraight lines inEuclidean space. For any pair of distinct non-antipodalpoints 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 theminor arc, and is the shortest surface-path between them. Itsarc length is thegreat-circle distance between the points (theintrinsic distance on a sphere), and is proportional to themeasure of thecentral 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. Anydiameter of any great circle coincides with a diameter of the sphere, and therefore every great circle isconcentric with the sphere and shares the sameradius. Any othercircle of the sphere is called asmall 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.

Thedisk bounded by a great circle is called agreat disk: it is the intersection of aball and a plane passing through its center.In higher dimensions, the great circles on then-sphere are the intersection of then-sphere with 2-planes that pass through the origin in theEuclidean spaceRn + 1.

Half of a great circle may be called agreatsemicircle (e.g., as in parts of ameridian in astronomy).

Derivation of shortest paths

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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 applycalculus of variations to it.

Consider the class of all regular paths from a pointp{\displaystyle p} to another pointq{\displaystyle q}. Introducespherical coordinates so thatp{\displaystyle p} 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

θ=θ(t),ϕ=ϕ(t),a≤t≤b{\displaystyle \theta =\theta (t),\quad \phi =\phi (t),\quad a\leq t\leq b}

providedϕ{\displaystyle \phi } is allowed to take on arbitrary real values. The infinitesimal arc length in these coordinates is

ds=rθ′2+ϕ′2sin2⁡θdt.{\displaystyle ds=r{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}\,dt.}

So the length of a curveγ{\displaystyle \gamma } fromp{\displaystyle p} toq{\displaystyle q} is afunctional of the curve given by

S[γ]=r∫abθ′2+ϕ′2sin2⁡θdt.{\displaystyle S[\gamma ]=r\int _{a}^{b}{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}\,dt.}

According to theEuler–Lagrange equation,S[γ]{\displaystyle S[\gamma ]} is minimized if and only if

sin2⁡θϕ′θ′2+ϕ′2sin2⁡θ=C{\displaystyle {\frac {\sin ^{2}\theta \phi '}{\sqrt {\theta '^{2}+\phi '^{2}\sin ^{2}\theta }}}=C},

whereC{\displaystyle C} is at{\displaystyle t}-independent constant, and

sin⁡θcos⁡θϕ′2θ′2+ϕ′2sin2⁡θ=ddtθ′θ′2+ϕ′2sin2⁡θ.{\displaystyle {\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 }}}.}

From the first equation of these two, it can be obtained that

ϕ′=Cθ′sin⁡θsin2⁡θ−C2{\displaystyle \phi '={\frac {C\theta '}{\sin \theta {\sqrt {\sin ^{2}\theta -C^{2}}}}}}.

Integrating both sides and considering the boundary condition, the real solution ofC{\displaystyle C} is zero. Thus,ϕ′=0{\displaystyle \phi '=0} andθ{\displaystyle \theta } can be any value between 0 andθ0{\displaystyle \theta _{0}}, indicating that the curve must lie on a meridian of the sphere. In aCartesian coordinate system, this is

xsin⁡ϕ0−ycos⁡ϕ0=0{\displaystyle x\sin \phi _{0}-y\cos \phi _{0}=0}

which is a plane through the origin, i.e., the center of the sphere.

Applications

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Some examples of great circles on thecelestial sphere include thecelestial horizon, thecelestial equator, and theecliptic. Great circles are also used as rather accurate approximations ofgeodesics on theEarth's surface for air or seanavigation (although itis not a perfect sphere), as well as on spheroidalcelestial bodies.

Theequator 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 theland and water hemispheres. A great circle divides the earth into twohemispheres and if a great circle passes through a point it must pass through itsantipodal point.

TheFunk transform integrates a function along all great circles of the sphere.

See also

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References

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  1. ^W., Weisstein, Eric."Great Circle -- from Wolfram MathWorld".mathworld.wolfram.com. Retrieved2022-09-30.{{cite web}}: CS1 maint: multiple names: authors list (link)
  2. ^Weintrit, Adam; Kopcz, Piotr (2014).Loxodrome (Rhumb Line), Orthodrome (Great Circle), Great Ellipse and Geodetic Line (Geodesic) in Navigation. USA: CRC Press, Inc.ISBN 978-1-138-00004-9.

External links

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