In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a great circle or orthodrome is the
circular
Circular may refer to:
* The shape of a circle
* ''Circular'' (album), a 2006 album by Spanish singer Vega
* Circular letter (disambiguation)
** Flyer (pamphlet), a form of advertisement
* Circular reasoning, a type of logical fallacy
* Circula ...
intersection
In mathematics, the intersection of two or more objects is another object consisting of everything that is contained in all of the objects simultaneously. For example, in Euclidean geometry, when two lines in a plane are not parallel, their i ...
of a
sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three ...
and a
plane
Plane(s) most often refers to:
* Aero- or airplane, a powered, fixed-wing aircraft
* Plane (geometry), a flat, 2-dimensional surface
Plane or planes may also refer to:
Biology
* Plane (tree) or ''Platanus'', wetland native plant
* Planes (gen ...
passing through the sphere's
center point.
Any
arc
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
of a great circle is a
geodesic
In geometry, a geodesic () is a curve representing in some sense the shortest path ( arc) between two points in a surface, or more generally in a Riemannian manifold. The term also has meaning in any differentiable manifold with a connection. ...
of the sphere, so that great circles in
spherical geometry
300px, A sphere with a spherical triangle on it.
Spherical geometry is the geometry of the two-dimensional surface of a sphere. In this context the word "sphere" refers only to the 2-dimensional surface and other terms like "ball" or "solid sp ...
are the natural analog of
straight lines
In geometry, a line is an infinitely long object with no width, depth, or curvature. Thus, lines are one-dimensional objects, though they may exist in two, three, or higher dimension spaces. The word ''line'' may also refer to a line segmen ...
in
Euclidean space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
. For any pair of distinct non-
antipodal point
Point or points may refer to:
Places
* Point, Lewis, a peninsula in the Outer Hebrides, Scotland
* Point, Texas, a city in Rains County, Texas, United States
* Point, the NE tip and a ferry terminal of Lismore, Inner Hebrides, Scotland
* 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
ARC may refer to:
Business
* Aircraft Radio Corporation, a major avionics manufacturer from the 1920s to the '50s
* Airlines Reporting Corporation, an airline-owned company that provides ticket distribution, reporting, and settlement services
* ...
is the
great-circle distance
The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle.
It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a ...
between the points (the
intrinsic distance on a sphere), and is proportional to the
measure
Measure may refer to:
* Measurement, the assignment of a number to a characteristic of an object or event
Law
* Ballot measure, proposed legislation in the United States
* Church of England Measure, legislation of the Church of England
* Mea ...
of the
central angle
A central angle is an angle whose apex (vertex) is the center O of a circle and whose legs (sides) are radii intersecting the circle in two distinct points A and B. Central angles are subtended by an arc between those two points, and the arc le ...
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
In geometry, a diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. It can also be defined as the longest chord of the circle. Both definitions are also valid for ...
of any great circle coincides with a diameter of the sphere, and therefore every great circle is
concentric
In geometry, two or more objects are said to be concentric, coaxal, or coaxial when they share the same center or axis. Circles, regular polygons and regular polyhedra, and spheres may be concentric to one another (sharing the same center point ...
with the sphere and shares the same
radius
In classical geometry, a radius ( : radii) of a circle or sphere is any of the line segments from its center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', meaning ray but also the ...
. Any other
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
Disk or disc may refer to:
* Disk (mathematics), a geometric shape
* Disk storage
Music
* Disc (band), an American experimental music band
* ''Disk'' (album), a 1995 EP by Moby
Other uses
* Disk (functional analysis), a subset of a vector sp ...
bounded by a great circle is called a ''great disk'': it is the intersection of a
ball
A ball is a round object (usually spherical, but can sometimes be ovoid) with several uses. It is used in ball games, where the play of the game follows the state of the ball as it is hit, kicked or thrown by players. Balls can also be used f ...
and a plane passing through its center.
In higher dimensions, the great circles on the
''n''-sphere are the intersection of the ''n''-sphere with 2-planes that pass through the origin in the Euclidean space .
Derivation of shortest paths
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
The calculus of variations (or Variational Calculus) is a field of mathematical analysis that uses variations, which are small changes in functions
and functionals, to find maxima and minima of functionals: mappings from a set of functions t ...
to it.
Consider the class of all regular paths from a point
to another point
. Introduce
spherical coordinates
In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the ''radial distance'' of that point from a fixed origin, its ''polar angle'' measu ...
so that
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
:
provided we allow
to take on arbitrary real values. The infinitesimal arc length in these coordinates is
:
So the length of a curve
from
to
is a
functional
Functional may refer to:
* Movements in architecture:
** Functionalism (architecture)
** Form follows function
* Functional group, combination of atoms within molecules
* Medical conditions without currently visible organic basis:
** Functional sy ...
of the curve given by
:
According to the
Euler–Lagrange equation
In the calculus of variations and classical mechanics, the Euler–Lagrange equations are a system of second-order ordinary differential equations whose solutions are stationary points of the given action functional. The equations were discovered ...
,