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A **great circle**, also known as an **orthodrome**, of a sphere is the intersection of the sphere and a plane that passes through the center point 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 all great circles have the same center and circumference as each other. This special case of a circle of a sphere is in opposition to a *small circle*, that is, the intersection of the sphere and a plane that does not pass through the center. Every circle in Euclidean 3-space is a great circle of exactly one sphere.

For most pairs of distinct points on the surface of a sphere, there is a unique great circle through the two points. The exception is a pair of antipodal points, for which there are infinitely many great circles. The minor arc of a great circle between two points is the shortest surface-path between them. In this sense, the minor arc is analogous to “straight lines” in Euclidean geometry. The length of the minor arc of a great circle is taken as the distance between two points on a surface of a sphere in Riemannian geometry where such great circles are called Riemannian circles. These great circles are the geodesics of the sphere.

The disk bounded by a great circle is called a *great disk*: it is the intersection of a ball 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 **R**^{n + 1}.

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 to another point . Introduce spherical coordinates 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 of the curve given by

According to the Euler–Lagrange equation, is minimized if and only if

- ,

where is a -independent constant, and

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

- .

Integrating both sides and considering the boundary condition, the real solution of is zero. Thus, and can be any value between 0 and , indicating that the curve must lie on a meridian of the sphere. In Cartesian coordinates, this is

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

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 the Earth's surface for air or sea navigation (although it 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 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.

- Great Circle – from MathWorld Great Circle description, figures, and equations. Mathworld, Wolfram Research, Inc. c1999
- Great Circles on Mercator's Chart by John Snyder with additional contributions by Jeff Bryant, Pratik Desai, and Carl Woll, Wolfram Demonstrations Project.
- Navigational Algorithms Paper: The Sailings.
- Chart Work - Navigational Algorithms Chart Work free software: Rhumb line, Great Circle, Composite sailing, Meridional parts. Lines of position Piloting - currents and coastal fix.

A **sphere** is a geometrical object in three-dimensional space that is the surface of a ball.

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* measured from a fixed zenith direction, and the *azimuthal angle* of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane. It can be seen as the three-dimensional version of the polar coordinate system.

In mathematics, an ** n-sphere** is a topological space that is homeomorphic to the set of points in (

In geometry, a **solid angle** is a measure of the amount of the field of view from some particular point that a given object covers. That is, it is a measure of how large the object appears to an observer looking from that point. The point from which the object is viewed is called the *apex* of the solid angle, and the object is said to *subtend* its solid angle from that point.

An **ellipsoid** is a surface that may be obtained from a sphere by deforming it by means of directional scalings, or more generally, of an affine transformation.

In mechanics and geometry, the **3D rotation group**, often denoted **SO(3)**, is the group of all rotations about the origin of three-dimensional Euclidean space under the operation of composition. By definition, a rotation about the origin is a transformation that preserves the origin, Euclidean distance, and orientation. Every non-trivial rotation is determined by its axis of rotation and its angle of rotation. Composing two rotations results in another rotation; every rotation has a unique inverse rotation; and the identity map satisfies the definition of a rotation. Owing to the above properties, the set of all rotations is a group under composition. Rotations are not commutative, making it a nonabelian group. Moreover, the rotation group has a natural structure as a manifold for which the group operations are smoothly differentiable; so it is in fact a Lie group. It is compact and has dimension 3.

**Elliptic geometry** is an example of a geometry in which Euclid's parallel postulate does not hold. Instead, as in spherical geometry, there are no parallel lines since any two lines must intersect. However, unlike in spherical geometry, two lines are usually assumed to intersect at a single point. Because of this, the elliptic geometry described in this article is sometimes referred to as *single elliptic geometry* whereas spherical geometry is sometimes referred to as *double elliptic geometry*.

The **great-circle distance** or **orthodromic distance** is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere. The distance between two points in Euclidean space is the length of a straight line between them, but on the sphere there are no straight lines. In spaces with curvature, straight lines are replaced by geodesics. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called *great circles*.

In non-Euclidean geometry, the **Poincaré half-plane model** is the upper half-plane, denoted below as **H**, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.

The **haversine formula** determines the great-circle distance between two points on a sphere given their longitudes and latitudes. Important in navigation, it is a special case of a more general formula in spherical trigonometry, the **law of haversines**, that relates the sides and angles of spherical triangles.

A **cone** is a three-dimensional geometric shape that tapers smoothly from a flat base to a point called the apex or vertex.

In geometry, **Euler's rotation theorem** states that, in three-dimensional space, any displacement of a rigid body such that a point on the rigid body remains fixed, is equivalent to a single rotation about some axis that runs through the fixed point. It also means that the composition of two rotations is also a rotation. Therefore the set of rotations has a group structure, known as a *rotation group*.

**Arc length** is the distance between two points along a section of a curve.

A **photon sphere** or **photon circle** is an area or region of space where gravity is so strong that photons are forced to travel in orbits. The radius of the photon sphere, which is also the lower bound for any stable orbit, is, for a Schwarzschild black hole:

The **Lambert azimuthal equal-area projection** is a particular mapping from a sphere to a disk. It accurately represents area in all regions of the sphere, but it does not accurately represent angles. It is named for the Swiss mathematician Johann Heinrich Lambert, who announced it in 1772. "Zenithal" being synonymous with "azimuthal", the projection is also known as the **Lambert zenithal equal-area projection**.

**Position angle**, usually abbreviated **PA**, is the convention for measuring angles on the sky in astronomy. The International Astronomical Union defines it as the angle measured relative to the north celestial pole (NCP), turning positive into the direction of the right ascension. In the standard (non-flipped) images this is a counterclockwise measure relative to the axis into the direction of positive declination.

In geometric topology, the **Clifford torus** is the simplest and most symmetric flat embedding of the cartesian product of two circles *S*^{1}_{a} and *S*^{1}_{b}. It is named after William Kingdon Clifford. It resides in **R**^{4}, as opposed to in **R**^{3}. To see why **R**^{4} is necessary, note that if *S*^{1}_{a} and *S*^{1}_{b} each exist in their own independent embedding spaces **R**^{2}_{a} and **R**^{2}_{b}, the resulting product space will be **R**^{4} rather than **R**^{3}. The historically popular view that the cartesian product of two circles is an **R**^{3} torus in contrast requires the highly asymmetric application of a rotation operator to the second circle, since that circle will only have one independent axis *z* available to it after the first circle consumes *x* and *y*.

A **great ellipse** is an ellipse passing through two points on a spheroid and having the same center as that of the spheroid. Equivalently, it is an ellipse on the surface of a spheroid and centered on the origin, or the curve formed by intersecting the spheroid by a plane through its center. For points that are separated by less than about a quarter of the circumference of the earth, about , the length of the great ellipse connecting the points is close to the geodesic distance. The great ellipse therefore is sometimes proposed as a suitable route for marine navigation. The great ellipse is special case of an earth section path.

**Geographical distance** is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. This distance is an element in solving the second (inverse) geodetic problem.

**Earth section paths** are paths on the earth defined by the intersection of a reference ellipsoid and a plane. Common examples of earth sections include the great ellipse and normal sections. This page provides a unifying approach to all earth sections and their associated geodetic problems.

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