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
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"). ...
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
geodesics. 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 points (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.
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
, 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
and
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
be their absolute differences; then
, 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 if one of the poles is used as an auxiliary third point on the sphere:
:
The problem is normally expressed in terms of finding the central angle
. 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
:
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 errors 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 is
numerically better-conditioned for small distances:
:
Historically, the use of this formula was simplified by the availability of tables for the
haversine function: hav(''θ'') = sin
2(''θ''/2).
Although this formula is accurate for most distances on a sphere, it too suffers from rounding errors for the special (and somewhat unusual) case of antipodal points. A formula that is accurate for all distances is the following special case of the
Vincenty formula for an ellipsoid with equal major and minor axes:
:
Here the quadrant for
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:
:
where
and
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
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 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,
, may be calculated as follows for the corresponding unit sphere, by means of
Cartesian subtraction:
:
The central angle is:
:
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
of 6378.137 km; distance
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
(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
, 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,
(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.
[
]
See also
*
Air navigation
*
Angular distance
*
Circumnavigation
Circumnavigation is the complete navigation around an entire island, continent, or astronomical body (e.g. a planet or moon). This article focuses on the circumnavigation of Earth.
The first recorded circumnavigation of the Earth was the ...
*
Flight planning
*
Geodesy
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equival ...
*
Geodesics on an ellipsoid
*
Geodetic system
A geodetic datum or geodetic system (also: geodetic reference datum, geodetic reference system, or geodetic reference frame) is a global datum reference or reference frame for precisely representing the position of locations on Earth or other pla ...
*
Geographical distance
*
Isoazimuthal
*
Loxodromic navigation
Loxodromic navigation (from Greek ''λοξóς'', oblique, and ''δρóμος'', path) is a method of navigation by following a rhumb line, a curve on the surface of the Earth that follows the same angle at the intersection with each meridian. This ...
*
Meridian arc
*
Rhumb line
*
Spherical geometry
*
Spherical trigonometry
References and notes
External links
GreatCircleat
MathWorld
{{DEFAULTSORT:Great-Circle Distance
Metric geometry
Spherical trigonometry
Distance
Spherical curves