Haversine formula

Last updated February 15, 2024

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.

These names follow from the fact that they are customarily written in terms of the haversine function, given by $hav(θ) = sin2(.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}θ/2)$ . The formulas could equally be written in terms of any multiple of the haversine, such as the older versine function (twice the haversine). Prior to the advent of computers, the elimination of division and multiplication by factors of two proved convenient enough that tables of haversine values and logarithms were included in 19th- and early 20th-century navigation and trigonometric texts.^[6]^[7]^[8] These days, the haversine form is also convenient in that it has no coefficient in front of the $sin 2$ function.

Formulation

Let the central angle $θ$ between any two points on a sphere be:

\theta ={\frac {d}{r}}

where:

$d$ is the distance between the two points along a great circle of the sphere (see spherical distance),
$r$ is the radius of the sphere.

The haversine formula allows the haversine of $θ$ (that is, $hav(θ)$ ) to be computed directly from the latitude (represented by $φ$ ) and longitude (represented by $λ$ ) of the two points:

\operatorname {hav} \left(\theta \right)=\operatorname {hav} \left(\varphi _{2}-\varphi _{1}\right)+\cos \left(\varphi _{1}\right)\cos \left(\varphi _{2}\right)\operatorname {hav} \left(\lambda _{2}-\lambda _{1}\right)

where

$φ 1$ , $φ 2$ are the latitude of point 1 and latitude of point 2,
$λ 1$ , $λ 2$ are the longitude of point 1 and longitude of point 2.

Finally, the haversine function $hav(θ)$ , applied above to both the central angle $θ$ and the differences in latitude and longitude, is

\operatorname {hav} (\theta )=\sin ^{2}\left({\frac {\theta }{2}}\right)={\frac {1-\cos(\theta )}{2}}

The haversine function computes half a versine of the angle $θ$ , or the squares of half chord of the angle on a unit circle (sphere).

To solve for the distance $d$ , apply the archaversine (inverse haversine) to $h = hav(θ)$ or use the arcsine (inverse sine) function:

d=r\operatorname {archav} (h)=2r\arcsin \left({\sqrt {h}}\right)

or more explicitly:

{\begin{aligned}d&=2r\arcsin \left({\sqrt {\operatorname {hav} (\varphi _{2}-\varphi _{1})+(1-\operatorname {hav} (\varphi _{1}-\varphi _{2})-\operatorname {hav} (\varphi _{1}+\varphi _{2}))\cdot \operatorname {hav} (\lambda _{2}-\lambda _{1})}}\right)\\&=2r\arcsin \left({\sqrt {\sin ^{2}\left({\frac {\varphi _{2}-\varphi _{1}}{2}}\right)+\left(1-\sin ^{2}\left({\frac {\varphi _{2}-\varphi _{1}}{2}}\right)-\sin ^{2}\left({\frac {\varphi _{2}+\varphi _{1}}{2}}\right)\right)\cdot \sin ^{2}\left({\frac {\lambda _{2}-\lambda _{1}}{2}}\right)}}\right)\\&=2r\arcsin \left({\sqrt {\sin ^{2}\left({\frac {\varphi _{2}-\varphi _{1}}{2}}\right)\cdot \cos ^{2}\left({\frac {\lambda _{2}-\lambda _{1}}{2}}\right)+\cos ^{2}\left({\frac {\varphi _{2}+\varphi _{1}}{2}}\right)\cdot \sin ^{2}\left({\frac {\lambda _{2}-\lambda _{1}}{2}}\right)}}\right)\\&=2r\arcsin \left({\sqrt {\sin ^{2}\left({\frac {\varphi _{2}-\varphi _{1}}{2}}\right)+\cos \varphi _{1}\cdot \cos \varphi _{2}\cdot \sin ^{2}\left({\frac {\lambda _{2}-\lambda _{1}}{2}}\right)}}\right).\end{aligned}}

^[9]

When using these formulae, one must ensure that $h$ does not exceed 1 due to a floating point error ( $d$ is only real for $0 \leq h \leq 1$ ). $h$ only approaches 1 for antipodal points (on opposite sides of the sphere)—in this region, relatively large numerical errors tend to arise in the formula when finite precision is used. Because $d$ is then large (approaching $π R$ , half the circumference) a small error is often not a major concern in this unusual case (although there are other great-circle distance formulas that avoid this problem). (The formula above is sometimes written in terms of the arctangent function, but this suffers from similar numerical problems near $h = 1$ .)

As described below, a similar formula can be written using cosines (sometimes called the spherical law of cosines, not to be confused with the law of cosines for plane geometry) instead of haversines, but if the two points are close together (e.g. a kilometer apart, on the Earth) one might end up with $cos(d / R) = 0.99999999$ , leading to an inaccurate answer. Since the haversine formula uses sines, it avoids that problem.

Either formula is only an approximation when applied to the Earth, which is not a perfect sphere: the "Earth radius" $R$ varies from 6356.752 km at the poles to 6378.137 km at the equator. More importantly, the radius of curvature of a north-south line on the earth's surface is 1% greater at the poles (≈6399.594 km) than at the equator (≈6335.439 km)—so the haversine formula and law of cosines cannot be guaranteed correct to better than 0.5%.^{[ citation needed ]} More accurate methods that consider the Earth's ellipticity are given by Vincenty's formulae and the other formulas in the geographical distance article.

The law of haversines

Given a unit sphere, a "triangle" on the surface of the sphere is defined by the great circles connecting three points $u$ , $v$ , and $w$ on the sphere. If the lengths of these three sides are $a$ (from $u$ to $v$ ), $b$ (from $u$ to $w$ ), and $c$ (from $v$ to $w$ ), and the angle of the corner opposite $c$ is $C$ , then the law of haversines states:^[10]

\operatorname {hav} (c)=\operatorname {hav} (a-b)+\sin(a)\sin(b)\operatorname {hav} (C).

Since this is a unit sphere, the lengths $a$ , $b$ , and $c$ are simply equal to the angles (in radians) subtended by those sides from the center of the sphere (for a non-unit sphere, each of these arc lengths is equal to its central angle multiplied by the radius $R$ of the sphere).

In order to obtain the haversine formula of the previous section from this law, one simply considers the special case where $u$ is the north pole, while $v$ and $w$ are the two points whose separation $d$ is to be determined. In that case, $a$ and $b$ are $π / 2 - φ 1,2$ (that is, the, co-latitudes), $C$ is the longitude separation $λ 2 - λ 1$ , and $c$ is the desired $d / R$ . Noting that $sin(π / 2 - φ) = cos(φ)$ , the haversine formula immediately follows.

To derive the law of haversines, one starts with the spherical law of cosines:

\cos(c)=\cos(a)\cos(b)+\sin(a)\sin(b)\cos(C).\,

As mentioned above, this formula is an ill-conditioned way of solving for $c$ when $c$ is small. Instead, we substitute the identity that $cos(θ) = 1 - 2 hav(θ)$ , and also employ the addition identity $cos(a - b) = cos(a) cos(b) + sin(a) sin(b)$ , to obtain the law of haversines, above.

Proof

One can prove the formula:

\operatorname {hav} \left(\theta \right)=\operatorname {hav} \left(\varphi _{2}-\varphi _{1}\right)+\cos \left(\varphi _{1}\right)\cos \left(\varphi _{2}\right)\operatorname {hav} \left(\lambda _{2}-\lambda _{1}\right)

by transforming the points given by their latitude and longitude into cartesian coordinates, then taking their dot product.

Consider two points ${\bf {p_{1},p_{2}}}$ on the unit sphere, given by their latitude $\varphi$ and longitude $\lambda$ :

{\begin{aligned}{\bf {p_{2}}}&=(\lambda _{2},\varphi _{2})\\{\bf {p_{1}}}&=(\lambda _{1},\varphi _{1})\end{aligned}}

These representations are very similar to spherical coordinates, however latitude is measured as angle from the equator and not the north pole. These points have the following representations in cartesian coordinates:

{\begin{aligned}{\bf {p_{2}}}&=(\cos(\lambda _{2})\cos(\varphi _{2}),\;\sin(\lambda _{2})\cos(\varphi _{2}),\;\sin(\varphi _{2}))\\{\bf {p_{1}}}&=(\cos(\lambda _{1})\cos(\varphi _{1}),\;\sin(\lambda _{1})\cos(\varphi _{1}),\;\sin(\varphi _{1}))\end{aligned}}

From here we could directly attempt to calculate the dot product and proceed, however the formulas become significantly simpler when we consider the following fact: the distance between the two points will not change if we rotate the sphere along the z-axis. This will in effect add a constant to $\lambda _{1},\lambda _{2}$ . Note that similar considerations do not apply to transforming the latitudes - adding a constant to the latitudes may change the distance between the points. By choosing our constant to be $-\lambda _{1}$ , and setting $\lambda '=\lambda _{2}-\lambda _{1}$ , our new points become:

{\begin{aligned}{\bf {p_{2}'}}&=(\cos(\lambda ')\cos(\varphi _{2}),\;\sin(\lambda ')\cos(\varphi _{2}),\;\sin(\varphi _{2}))\\{\bf {p_{1}'}}&=(\cos(0)\cos(\varphi _{1}),\;\sin(0)\cos(\varphi _{1}),\;\sin(\varphi _{1}))\\&=(\cos(\varphi _{1}),\;0,\;\sin(\varphi _{1}))\end{aligned}}

With $\theta$ denoting the angle between ${\bf {p_{1}}}$ and ${\bf {p_{2}}}$ , we now have that:

{\begin{aligned}\cos(\theta )&=\langle {\bf {p_{1}}},{\bf {p_{2}}}\rangle =\langle {\bf {p_{1}'}},{\bf {p_{2}'}}\rangle =\cos(\lambda ')\cos(\varphi _{1})\cos(\varphi _{2})+\sin(\varphi _{1})\sin(\varphi _{2})\\&=\sin(\varphi _{2})\sin(\varphi _{1})+\cos(\varphi _{2})\cos(\varphi _{1})-\cos(\varphi _{2})\cos(\varphi _{1})+\cos(\lambda ')\cos(\varphi _{2})\cos(\varphi _{1})\\&=\cos(\varphi _{2}-\varphi _{1})+\cos(\varphi _{2})\cos(\varphi _{1})(-1+\cos(\lambda '))\Rightarrow \\\operatorname {hav} \left(\theta \right)&=\operatorname {hav} \left(\varphi _{2}-\varphi _{1}\right)+\cos(\varphi _{2})\cos(\varphi _{1})\operatorname {hav} \left(\lambda _{2}-\lambda _{1}\right)\end{aligned}}

Related Research Articles

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a given point in space is specified by three numbers, : the radial distance of the radial liner connecting the point to the fixed point of origin ; the polar angle θ of the radial line r; and the azimuthal angle φ of the radial line r.

In mathematics, an $n$ -sphere or hypersphere is an $n$ -dimensional generalization of the 1-dimensional circle and 2-dimensional sphere to any non-negative integer $n$ . The $n$ -sphere is the setting for $n$ -dimensional spherical geometry.

<span class="mw-page-title-main">Ellipsoid</span> Quadric surface that looks like a deformed sphere

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

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. A list of the spherical harmonics is available in Table of spherical harmonics.

The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle.

In probability theory, the Borel–Kolmogorov paradox is a paradox relating to conditional probability with respect to an event of probability zero. It is named after Émile Borel and Andrey Kolmogorov.

Orthographic projection in cartography has been used since antiquity. Like the stereographic projection and gnomonic projection, orthographic projection is a perspective projection in which the sphere is projected onto a tangent plane or secant plane. The point of perspective for the orthographic projection is at infinite distance. It depicts a hemisphere of the globe as it appears from outer space, where the horizon is a great circle. The shapes and areas are distorted, particularly near the edges.

The Mollweide projection is an equal-area, pseudocylindrical map projection generally used for maps of the world or celestial sphere. It is also known as the Babinet projection, homalographic projection, homolographic projection, and elliptical projection. The projection trades accuracy of angle and shape for accuracy of proportions in area, and as such is used where that property is needed, such as maps depicting global distributions.

In cartography, a Tissot's indicatrix is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two principal directions along which scale is maximal and minimal at that point on the map.

<span class="mw-page-title-main">Great-circle navigation</span> Flight or sailing route along the shortest path between two points on a globes surface

Great-circle navigation or orthodromic navigation is the practice of navigating a vessel along a great circle. Such routes yield the shortest distance between two points on the globe.

The Cassini projection is a map projection first described in an approximate form by César-François Cassini de Thury in 1745. Its precise formulas were found through later analysis by Johann Georg von Soldner around 1810. It is the transverse aspect of the equirectangular projection, in that the globe is first rotated so the central meridian becomes the "equator", and then the normal equirectangular projection is applied. Considering the earth as a sphere, the projection is composed of the operations:

The General Perspective projection is a map projection. When the Earth is photographed from space, the camera records the view as a perspective projection. When the camera is aimed toward the center of the Earth, the resulting projection is called Vertical Perspective. When aimed in other directions, the resulting projection is called a Tilted Perspective.

In physics and mathematics, the solid harmonics are solutions of the Laplace equation in spherical polar coordinates, assumed to be (smooth) functions $. There are two kinds: the regular solid harmonics, which are well-defined at the origin and the irregular solid harmonics, which are singular at the origin. Both sets of functions play an important role in potential theory, and are obtained by rescaling spherical harmonics appropriately:$

The goat grazing problem is either of two related problems in recreational mathematics involving a tethered goat grazing a circular area: the interior grazing problem and the exterior grazing problem. The former involves grazing the interior of a circular area, and the latter, grazing an exterior of a circular area. For the exterior problem, the constraint that the rope can not enter the circular area dictates that the grazing area forms an involute. If the goat were instead tethered to a post on the edge of a circular path of pavement that did not obstruct the goat, the interior and exterior problem would be complements of a simple circular area.

In fluid dynamics, the Oseen equations describe the flow of a viscous and incompressible fluid at small Reynolds numbers, as formulated by Carl Wilhelm Oseen in 1910. Oseen flow is an improved description of these flows, as compared to Stokes flow, with the (partial) inclusion of convective acceleration.

<span class="mw-page-title-main">Geographical distance</span> Distance measured along the surface of the Earth

Geographical distance or geodetic distance is the distance measured along the surface of the Earth, or the shortest arch length.

The Eckert IV projection is an equal-area pseudocylindrical map projection. The length of the polar lines is half that of the equator, and lines of longitude are semiellipses, or portions of ellipses. It was first described by Max Eckert in 1906 as one of a series of three pairs of pseudocylindrical projections. Within each pair, meridians are the same whereas parallels differ. Odd-numbered projections have parallels spaced equally, whereas even-numbered projections have parallels spaced to preserve area. Eckert IV is paired with Eckert III.

In Einstein's theory of general relativity, the interior Schwarzschild metric is an exact solution for the gravitational field in the interior of a non-rotating spherical body which consists of an incompressible fluid and has zero pressure at the surface. This is a static solution, meaning that it does not change over time. It was discovered by Karl Schwarzschild in 1916, who earlier had found the exterior Schwarzschild metric.

The Strebe 1995 projection, Strebe projection, Strebe lenticular equal-area projection, or Strebe equal-area polyconic projection is an equal-area map projection presented by Daniel "daan" Strebe in 1994. Strebe designed the projection to keep all areas proportionally correct in size; to push as much of the inevitable distortion as feasible away from the continental masses and into the Pacific Ocean; to keep a familiar equatorial orientation; and to do all this without slicing up the map.

References

↑ van Brummelen, Glen Robert (2013). Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. Princeton University Press. ISBN 9780691148922. 0691148929. Retrieved 2015-11-10.
↑ de Mendoza y Ríos, Joseph (1795). Memoria sobre algunos métodos nuevos de calcular la longitud por las distancias lunares: y aplicacion de su teórica á la solucion de otros problemas de navegacion (in Spanish). Madrid, Spain: Imprenta Real.
↑ Cajori, Florian (1952) [1929]. A History of Mathematical Notations. Vol. 2 (2 (3rd corrected printing of 1929 issue) ed.). Chicago: Open court publishing company. p. 172. ISBN 978-1-60206-714-1. 1602067147. Retrieved 2015-11-11. The haversine first appears in the tables of logarithmic versines of José de Mendoza y Rios (Madrid, 1801, also 1805, 1809), and later in a treatise on navigation of James Inman (1821). (NB. ISBN and link for reprint of second edition by Cosimo, Inc., New York, 2013.)
↑ Inman, James (1835) [1821]. Navigation and Nautical Astronomy: For the Use of British Seamen (3 ed.). London, UK: W. Woodward, C. & J. Rivington. Retrieved 2015-11-09. (Fourth edition: .)
↑ "haversine". Oxford English Dictionary (2nd ed.). Oxford University Press. 1989.
↑ H. B. Goodwin, The haversine in nautical astronomy, Naval Institute Proceedings, vol. 36, no. 3 (1910), pp. 735–746: Evidently if a Table of Haversines is employed we shall be saved in the first instance the trouble of dividing the sum of the logarithms by two, and in the second place of multiplying the angle taken from the tables by the same number. This is the special advantage of the form of table first introduced by Professor Inman, of the Portsmouth Royal Navy College, nearly a century ago.
↑ W. W. Sheppard and C. C. Soule, Practical navigation (World Technical Institute: Jersey City, 1922).
↑ E. R. Hedrick, Logarithmic and Trigonometric Tables (Macmillan, New York, 1913).
↑ Gade, Kenneth (2010). "A Non-singular Horizontal Position Representation". Journal of Navigation. 63 (3): 395–417. Bibcode:2010JNav...63..395G. doi:10.1017/S0373463309990415. ISSN 0373-4633.
↑ Korn, Grandino Arthur; Korn, Theresa M. (2000) [1922]. "Appendix B: B9. Plane and Spherical Trigonometry: Formulas Expressed in Terms of the Haversine Function". Mathematical handbook for scientists and engineers: Definitions, theorems, and formulas for reference and review (3rd ed.). Mineola, New York: Dover Publications. pp. 892–893. ISBN 978-0-486-41147-7.

External links

Implementations of the haversine formula in 91 languages at rosettacode.org and in 17 languages on codecodex.com Archived 2018-08-14 at the Wayback Machine
Other implementations in C++, C (MacOS), Pascal Archived 2019-01-16 at the Wayback Machine , Python, Ruby, JavaScript, PHP Archived 2018-08-12 at the Wayback Machine ,Matlab Archived 2020-05-13 at the Wayback Machine , MySQL

This page is based on this Wikipedia article
Text is available under the CC BY-SA 4.0 license; additional terms may apply.
Images, videos and audio are available under their respective licenses.

[Brummelen_2013-1] van Brummelen, Glen Robert (2013). Heavenly Mathematics: The Forgotten Art of Spherical Trigonometry. Princeton University Press. ISBN 9780691148922. 0691148929. Retrieved 2015-11-10.

[Ríos_1795-2] Mendoza y Ríos, Joseph (1795). Memoria sobre algunos métodos nuevos de calcular la longitud por las distancias lunares: y aplicacion de su teórica á la solucion de otros problemas de navegacion (in Spanish). Madrid, Spain: Imprenta Real.

[Cajori_1929-3] Cajori, Florian (1952) [1929]. A History of Mathematical Notations. Vol. 2 (2 (3rd corrected printing of 1929 issue) ed.). Chicago: Open court publishing company. p. 172. ISBN 978-1-60206-714-1. 1602067147. Retrieved 2015-11-11. The haversine first appears in the tables of logarithmic versines of José de Mendoza y Rios (Madrid, 1801, also 1805, 1809), and later in a treatise on navigation of James Inman (1821). (NB. ISBN and link for reprint of second edition by Cosimo, Inc., New York, 2013.)

[Inman_1835-4] Inman, James (1835) [1821]. Navigation and Nautical Astronomy: For the Use of British Seamen (3 ed.). London, UK: W. Woodward, C. & J. Rivington. Retrieved 2015-11-09. (Fourth edition: .)

[OED_1989_Haversine-5] "haversine". Oxford English Dictionary (2nd ed.). Oxford University Press. 1989.

[6] H. B. Goodwin, The haversine in nautical astronomy, Naval Institute Proceedings, vol. 36, no. 3 (1910), pp. 735–746: Evidently if a Table of Haversines is employed we shall be saved in the first instance the trouble of dividing the sum of the logarithms by two, and in the second place of multiplying the angle taken from the tables by the same number. This is the special advantage of the form of table first introduced by Professor Inman, of the Portsmouth Royal Navy College, nearly a century ago.

[7] W. W. Sheppard and C. C. Soule, Practical navigation (World Technical Institute: Jersey City, 1922).

[8] E. R. Hedrick, Logarithmic and Trigonometric Tables (Macmillan, New York, 1913).

[Gade2010-9] Gade, Kenneth (2010). "A Non-singular Horizontal Position Representation". Journal of Navigation. 63 (3): 395–417. Bibcode:2010JNav...63..395G. doi:10.1017/S0373463309990415. ISSN 0373-4633.

[Korn_2000-10] Korn, Grandino Arthur; Korn, Theresa M. (2000) [1922]. "Appendix B: B9. Plane and Spherical Trigonometry: Formulas Expressed in Terms of the Haversine Function". Mathematical handbook for scientists and engineers: Definitions, theorems, and formulas for reference and review (3rd ed.). Mineola, New York: Dover Publications. pp. 892–893. ISBN 978-0-486-41147-7.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

[10]