The great-circle distance, orthodromic distance, or spherical distance is the

distance Distance is a numerical or occasionally qualitative measurement of how far apart objects or points are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two counties over"). ...

along a

great circle In mathematics, a great circle or orthodrome is the circular intersection of a sphere and a plane passing through the sphere's center point. Any arc of a great circle is a geodesic of the sphere, so that great circles in spherical geome ...

. It is the shortest

between two

points 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 * Points ...

on the surface of a

sphere A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the c ...

, measured along the surface of the sphere (as opposed to a straight line through the sphere's interior). The distance between two points in

Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidea ...

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

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 connecti ...

s. Geodesics on the sphere are circles on the sphere whose centers coincide with the center of the sphere, and are called 'great circles'. The determination of the great-circle distance is part of the more general problem of

great-circle navigation Great-circle navigation or orthodromic navigation (related to orthodromic course; from the Greek ''ορθóς'', right angle, and ''δρóμος'', path) is the practice of navigating a vessel (a ship or aircraft) along a great circle. Such rout ...

, which also computes the azimuths at the end points and intermediate way-points. Through any two points on a sphere that are not

antipodal point In mathematics, antipodal points of a sphere are those diametrically opposite to each other (the specific qualities of such a definition are that a line drawn from the one to the other passes through the center of the sphere so forms a true d ...

s (directly opposite each other), there is a unique great circle. The two points separate the great circle into two arcs. The length of the shorter arc is the great-circle distance between the points. A great circle endowed with such a distance is called a Riemannian circle in

Riemannian geometry Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point ...

. Between antipodal points, there are infinitely many great circles, and all great circle arcs between antipodal points have a length of half the

circumference In geometry, the circumference (from Latin ''circumferens'', meaning "carrying around") is the perimeter of a circle or ellipse. That is, the circumference would be the arc length of the circle, as if it were opened up and straightened out ...

of the circle, or

\pi r

, where ''r'' is the

radius In classical geometry, a radius (plural, : radii) of a circle or sphere is any of the line segments from its Centre (geometry), center to its perimeter, and in more modern usage, it is also their length. The name comes from the latin ''radius'', ...

of the sphere. The

Earth Earth is the third planet from the Sun and the only astronomical object known to harbor life. While large volumes of water can be found throughout the Solar System, only Earth sustains liquid surface water. About 71% of Earth's sur ...

is nearly spherical, so great-circle distance formulas give the distance between points on the surface of the Earth correct to within about 0.5%. The

vertex Vertex, vertices or vertexes may refer to: Science and technology Mathematics and computer science *Vertex (geometry), a point where two or more curves, lines, or edges meet *Vertex (computer graphics), a data structure that describes the position ...

is the highest-latitude point on a great circle.

Formulae

Let

\lambda_1, \phi_1

and

\lambda_2, \phi_2

be the geographical

longitude Longitude (, ) is a geographic coordinate that specifies the east– west position of a point on the surface of the Earth, or another celestial body. It is an angular measurement, usually expressed in degrees and denoted by the Greek let ...

and

latitude In geography, latitude is a coordinate that specifies the north– south position of a point on the surface of the Earth or another celestial body. Latitude is given as an angle that ranges from –90° at the south pole to 90° at the north ...

of two points 1 and 2, and

\Delta\lambda, \Delta\phi

be their absolute differences; then

\Delta\sigma

, 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 ...

between them, is given by the

spherical law of cosines In spherical trigonometry, the law of cosines (also called the cosine rule for sides) is a theorem relating the sides and angles of spherical triangles, analogous to the ordinary law of cosines from plane trigonometry. Given a unit sphere, a "sph ...

if one of the poles is used as an auxiliary third point on the sphere: :

\Delta\sigma = \arccos\bigl(\sin\phi_1\sin\phi_2 + \cos\phi_1\cos\phi_2\cos(\Delta\lambda)\bigr).

The problem is normally expressed in terms of finding the central angle

\Delta\sigma

. Given this angle in radians, the actual

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 * ...

''d'' on a sphere of radius ''r'' can be trivially computed as :

d = r \, \Delta\sigma.

Computational formulas

On computer systems with low

floating point In computing, floating-point arithmetic (FP) is arithmetic that represents real numbers approximately, using an integer with a fixed precision, called the significand, scaled by an integer exponent of a fixed base. For example, 12.345 can ...

precision, the spherical law of cosines formula can have large

rounding error A roundoff error, also called rounding error, is the difference between the result produced by a given algorithm using exact arithmetic and the result produced by the same algorithm using finite-precision, rounded arithmetic. Rounding errors are d ...

s if the distance is small (if the two points are a kilometer apart on the surface of the Earth, the cosine of the central angle is near 0.99999999). For modern 64-bit floating-point numbers, the spherical law of cosines formula, given above, does not have serious rounding errors for distances larger than a few meters on the surface of the Earth. The

haversine formula 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, ...

is numerically better-conditioned for small distances: :

\begin
  \Delta\sigma &= \operatorname\left( \operatorname\left(\Delta\phi\right) + \left(1 - \operatorname(\Delta\phi) - \operatorname(\phi_1 + \phi_2)\right)\cdot\operatorname\left(\Delta\lambda\right)\right) \\
               &= 2\arcsin \sqrt.
\end

Historically, the use of this formula was simplified by the availability of tables for the

haversine The versine or versed sine is a trigonometric function found in some of the earliest (Sanskrit ''Aryabhatia'',Vincenty formula for an ellipsoid with equal major and minor axes: :

\Delta\sigma =
  \arctan \frac
     .

Here the quadrant for

\Delta\sigma

should be governed by the signs of the numerator and denominator of the right hand side, e.g., using the

atan2 In computing and mathematics, the function atan2 is the 2-argument arctangent. By definition, \theta = \operatorname(y, x) is the angle measure (in radians, with -\pi < \theta \leq \pi) between the positive

function.

Vector version

Another representation of similar formulas, but using normal vectors instead of latitude and longitude to describe the positions, is found by means of 3D vector algebra, using the

dot product In mathematics, the dot product or scalar productThe term ''scalar product'' means literally "product with a scalar as a result". It is also used sometimes for other symmetric bilinear forms, for example in a pseudo-Euclidean space. is an alg ...

cross product In mathematics, the cross product or vector product (occasionally directed area product, to emphasize its geometric significance) is a binary operation on two vectors in a three-dimensional oriented Euclidean vector space (named here E), and ...

, or a combination: :

\begin
  \Delta\sigma &= \arccos \left(\mathbf n_1 \cdot \mathbf n_2\right) \\
               &= \arcsin \left,  \mathbf n_1 \times \mathbf n_2 \ \\
               &= \arctan \frac \\
\end

where

\mathbf n_1

and

\mathbf n_2

are the normals to the ellipsoid at the two positions 1 and 2. Similarly to the equations above based on latitude and longitude, the expression based on arctan is the only one that is well-conditioned for all angles. The expression based on arctan requires the magnitude of the cross product over the dot product.

From chord length

A line through three-dimensional space between points of interest on a

spherical Earth Spherical Earth or Earth's curvature refers to the approximation of figure of the Earth as a sphere. The earliest documented mention of the concept dates from around the 5th century BC, when it appears in the writings of Greek philosophers. ...

is the chord of the great circle between the points. The

between the two points can be determined from the chord length. The great circle distance is proportional to the central angle. The great circle chord length,

C_h\,\!

, may be calculated as follows for the corresponding unit sphere, by means of Cartesian subtraction: :

\begin
  \Delta &= \cos\phi_2\cos\lambda_2 - \cos\phi_1\cos\lambda_1;\\
  \Delta &= \cos\phi_2\sin\lambda_2 - \cos\phi_1\sin\lambda_1;\\
  \Delta &= \sin\phi_2 - \sin\phi_1;\\
          C &= \sqrt
\end

The central angle is: :

\Delta\sigma=2\arcsin \frac .

Radius for spherical Earth

The shape of the Earth closely resembles a flattened sphere (a

spheroid A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has ...

) with equatorial radius

a

of 6378.137 km; distance

b

from the center of the spheroid to each pole is 6356.7523142 km. When calculating the length of a short north-south line at the equator, the circle that best approximates that line has a radius of

\frac

(which equals the meridian's

semi-latus rectum In mathematics, a conic section, quadratic curve or conic is a curve obtained as the intersection of the surface of a cone with a plane. The three types of conic section are the hyperbola, the parabola, and the ellipse; the circle is a sp ...

), or 6335.439 km, while the spheroid at the poles is best approximated by a sphere of radius

\frac

, or 6399.594 km, a 1% difference. So long as a spherical Earth is assumed, any single formula for distance on the Earth is only guaranteed correct within 0.5% (though better accuracy is possible if the formula is only intended to apply to a limited area). Using the mean earth radius,

R_1 = \frac(2a + b) \approx 6371.009\text

(for the

WGS84 The World Geodetic System (WGS) is a standard used in cartography, geodesy, and satellite navigation including GPS. The current version, WGS 84, defines an Earth-centered, Earth-fixed coordinate system and a geodetic datum, and also desc ...

ellipsoid) means that in the limit of small flattening, the mean square

relative error The approximation error in a data value is the discrepancy between an exact value and some ''approximation'' to it. This error can be expressed as an absolute error (the numerical amount of the discrepancy) or as a relative error (the absolute er ...

in the estimates for distance is minimized.

References and notes

External links

GreatCircle
at

MathWorld ''MathWorld'' is an online mathematics reference work, created and largely written by Eric W. Weisstein. It is sponsored by and licensed to Wolfram Research, Inc. and was partially funded by the National Science Foundation's National Science Di ...

{{DEFAULTSORT:Great-Circle Distance Metric geometry Spherical trigonometry Distance Spherical curves