haversine formula

Let the central angle {{math|θ}} between any two points on a sphere be:

:$\backslash theta\; =\; \backslash frac\{d\}\{r\}$

where

• {{math|d}} is the distance between the two points along a great circle of the sphere (see spherical distance),
• {{math|r}} is the radius of the sphere.

The haversine formula allows the haversine of {{math|θ}} to be computed directly from the latitude (represented by {{math|φ}}) and longitude (represented by {{math|λ}}) of the two points:

:$$

\operatorname{hav}\theta =

\operatorname{hav}\left(\Delta \varphi \right) + \cos\left(\varphi_1\right)\cos\left(\varphi_2\right)\operatorname{hav}\left(\Delta \lambda \right)

where

• {{math|φ₁}}, {{math|φ₂}} are the latitude of point 1 and latitude of point 2,
• {{math|λ₁}}, {{math|λ₂}} are the longitude of point 1 and longitude of point 2,
• $\backslash Delta\; \backslash varphi\; =\; \backslash varphi\_2\; -\; \backslash varphi\_1$, $\backslash Delta\; \backslash lambda\; =\; \backslash lambda\_2\; -\; \backslash lambda\_1$.

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

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

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

To solve for the distance {{math|d}}, apply the archaversine (inverse haversine) to {{math|hav(θ)}} or use the arcsine (inverse sine) function:

: $d\; =\; r\backslash operatorname\{archav\}(\backslash operatorname\{hav\}\backslash theta)\; =\; 2r\backslash arcsin\backslash left(\backslash sqrt\{\backslash operatorname\{hav\}\backslash theta\}\backslash right)$

or more explicitly:

: $\backslash begin\{align\}$

d &= 2r \arcsin\left(\sqrt{\operatorname{hav}(\Delta \varphi ) + ( 1 - \operatorname{hav}(\Delta \varphi) - \operatorname{hav}(2 \varphi_\text{m} ))\cdot\operatorname{hav}(\Delta \lambda)}\right) \\

&= 2r \arcsin\left(\sqrt{\sin^2\left(\frac{\Delta \varphi }{2}\right) + \left(1- \sin^2\left(\frac{\Delta \varphi }{2}\right) - \sin^2\left(\varphi_\text{m}\right)\right) \cdot \sin^2\left(\frac{\Delta \lambda}{2}\right)}\right) \\

&= 2r \arcsin\left(\sqrt{\sin^2\left(\frac{\Delta \varphi }{2}\right) + \cos \varphi_1 \cdot \cos \varphi_2 \cdot \sin^2\left(\frac{\Delta \lambda}{2}\right)}\right) \\

&= 2r \arcsin\left(\sqrt{\sin^2\left(\frac{\Delta \varphi }{2}\right) \cdot \cos^2\left(\frac{\Delta \lambda}{2}\right) + \cos^2\left(\varphi_\text{m}\right) \cdot \sin^2\left(\frac{\Delta \lambda}{2}\right)}\right) \\

&= 2r \arcsin\left(\sqrt{\frac{1 - \cos\left(\Delta \varphi \right) + \cos \varphi_1 \cdot \cos \varphi_2 \cdot \left(1 - \cos\left(\Delta \lambda\right)\right)}{2}}\right)

\end{align}{{cite journal|last1=Gade|first1=Kenneth|title=A Non-singular Horizontal Position Representation|journal=Journal of Navigation|volume=63|issue=3|year=2010|pages=395–417|issn=0373-4633|doi=10.1017/S0373463309990415|bibcode=2010JNav...63..395G }}

where

$\backslash varphi\_\backslash text\{m\}\; =\; \backslash frac\{\backslash varphi\_2\; +\; \backslash varphi\_1\}\{2\}$.

When using these formulae, one must ensure that {{math|h {{=}} hav(θ)}} does not exceed 1 due to a floating point error ({{math|d}} is real only for {{math|0 ≤ h ≤ 1}}). {{math|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 {{math|d}} is then large (approaching {{math|πR}}, 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 {{math|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 {{math|cos({{sfrac|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" {{math|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|reason=Your explanation here|date=January 2019}} More accurate methods that consider the Earth's ellipticity are given by Vincenty's formulae and the other formulas in the geographical distance article.

File:Law-of-haversines.svg

Given a unit sphere, a "triangle" on the surface of the sphere is defined by the great circles connecting three points {{math|u}}, {{math|v}}, and {{math|w}} on the sphere. If the lengths of these three sides are {{math|a}} (from {{math|u}} to {{math|v}}), {{math|b}} (from {{math|u}} to {{math|w}}), and {{math|c}} (from {{math|v}} to {{math|w}}), and the angle of the corner opposite {{math|c}} is {{math|C}}, then the law of haversines states:{{cite book |title=Mathematical handbook for scientists and engineers: Definitions, theorems, and formulas for reference and review |first1=Grandino Arthur |last1=Korn |first2=Theresa M.|author2-link= Theresa M. Korn |last2=Korn |edition=3rd |date=2000 |orig-year=1922 |publisher=Dover Publications |location=Mineola, New York |chapter=Appendix B: B9. Plane and Spherical Trigonometry: Formulas Expressed in Terms of the Haversine Function |pages=892–893 |isbn=978-0-486-41147-7}}

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

Since this is a unit sphere, the lengths {{math|a}}, {{math|b}}, and {{math|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 {{math|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 {{math|u}} is the north pole, while {{math|v}} and {{math|w}} are the two points whose separation {{math|d}} is to be determined. In that case, {{math|a}} and {{math|b}} are {{math|{{sfrac|π|2}} − φ_1,2}} (that is, the, co-latitudes), {{math|C}} is the longitude separation {{math|λ₂ − λ₁}}, and {{math|c}} is the desired {{math|{{sfrac|d|R}}}}. Noting that {{math|sin({{sfrac|π|2}} − φ) {{=}} cos(φ)}}, the haversine formula immediately follows.

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

:$\backslash cos(c)\; =\; \backslash cos(a)\backslash cos(b)\; +\; \backslash sin(a)\backslash sin(b)\backslash cos(C).\; \backslash ,$

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

One can prove the formula:

:$$

\operatorname{hav}\left(\theta\right) =

\operatorname{hav}\left(\Delta \varphi \right) + \cos\left(\varphi_1\right)\cos\left(\varphi_2\right)\operatorname{hav}\left(\Delta \lambda\right)

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

Consider two points $\backslash bf\; p\_1,p\_2$ on the unit sphere, given by their latitude $\backslash varphi$ and longitude $\backslash lambda$:

:$\backslash begin\{align\}$

{\bf p_2} &= (\lambda_2, \varphi_2) \\

{\bf p_1} &= (\lambda_1, \varphi_1)

\end{align}

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:

:$\backslash begin\{align\}$

{\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{align}

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 $\backslash lambda\_1,\; \backslash lambda\_2$. 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 $-\backslash lambda\_1$, and setting $\backslash lambda\text{'}\; =\; \backslash Delta\; \backslash lambda$, our new points become:

:$\backslash begin\{align\}$

{\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{align}

With $\backslash theta$ denoting the angle between $\{\backslash bf\; p\_1\}$ and $\{\backslash bf\; p\_2\}$, we now have that:

:$\backslash begin\{align\}$

\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(\Delta \varphi) + \cos(\varphi_2)\cos(\varphi_1)(-1 + \cos(\lambda')) \Rightarrow \\

\operatorname{hav}\left(\theta\right)

&= \operatorname{hav}\left(\Delta \varphi \right) + \cos(\varphi_2)\cos(\varphi_1)\operatorname{hav}\left(\lambda' \right)

\end{align}

File:Geodesic from the White House to the Eiffel Tower.svg, with geodesic circles (blue) at multiples of 1,000 km distance from Washington.]]

The haversine formula can be used to find the approximate distance between the White House in Washington, D.C. (latitude 38.898° N, longitude 77.037° W) and the Eiffel Tower in Paris (latitude 48.858° N, longitude 2.294° E). The difference in latitudes is $\backslash Delta\; \backslash varphi\; =\; \{\}$9.96° and the difference in longitudes is $\backslash Delta\backslash lambda\; =\; \{\}$79.331°. Inputting these into the haversine formula,

$$\backslash begin\{align\}$$

\operatorname{hav}\left(\theta\right)

&= \operatorname{hav}(\Delta \varphi ) + \cos(\varphi_1)\cos(\varphi_2)\operatorname{hav}(\Delta \lambda) \\[5mu]

&= \operatorname{hav}(9.96^\circ) + \cos(38.898^\circ)\cos(48.858^\circ)\operatorname{hav}(79.331^\circ) \\[5mu]

&\approx 0.0075356 + 0.77827 \times 0.65793 \times 0.40743 \\[5mu]

&\approx 0.21616 \\[5mu]

\theta &\approx 55.411^\circ.

\end{align}

The great-circle distance is this central angle, in radians, multiplied by the average radius of the Earth,

$$0.96710\; \backslash times\; 6371.2\backslash \; \backslash text\{km\}\; \backslash approx\; 6161.6\backslash \; \backslash text\{km\}.$$

By comparison, using a more accurate ellipsoidal model of the earth, the geodesic distance between these landmarks can be computed as approximately 6177.45 km.This calculation was made using the open-source geodesic calculation software [https://geographiclib.sourceforge.io GeographicLib], assuming the WGS84 ellipsoid. See {{cite journal |first=Charles F. F. |last=Karney |title=Algorithms for geodesics |journal=Journal of Geodesy |volume=87 |number=1 |pages=43–55 |year=2013 |doi=10.1007/s00190-012-0578-z |doi-access=free|arxiv=1109.4448 }}

• Sight reduction
• Vincenty's formulae
• Cosine distance

• U. S. Census Bureau Geographic Information Systems FAQ, (content has been moved to [http://www.movable-type.co.uk/scripts/GIS-FAQ-5.1.html What is the best way to calculate the distance between 2 points?])
• R. W. Sinnott, "Virtues of the Haversine", Sky and Telescope 68 (2), 159 (1984).
• {{cite web|url=http://mathforum.org/library/drmath/view/51879.html |title=Deriving the haversine formula|work=Ask Dr. Math|date=April 20–21, 1999|archive-url=https://web.archive.org/web/20200120134215/http://mathforum.org/library/drmath/view/51879.html|archive-date=20 January 2020|url-status=dead}}
• W. Gellert, S. Gottwald, M. Hellwich, H. Kästner, and H. Küstner, The VNR Concise Encyclopedia of Mathematics, 2nd ed., ch. 12 (Van Nostrand Reinhold: New York, 1989).

• Implementations of the haversine formula [http://rosettacode.org/wiki/Haversine_formula in 91 languages at rosettacode.org] and [http://www.codecodex.com/wiki/Calculate_Distance_Between_Two_Points_on_a_Globe in 17 languages on codecodex.com] {{Webarchive|url=https://web.archive.org/web/20180814170246/http://www.codecodex.com/wiki/Calculate_Distance_Between_Two_Points_on_a_Globe |date=2018-08-14 }}
• Other implementations in [http://blog.julien.cayzac.name/2008/10/arc-and-distance-between-two-points-on.html C++], [http://www.jaimerios.com/?p=39 C (MacOS)], [http://scifunam.fisica.unam.mx/mir/codes.html#haversine Pascal] {{Webarchive|url=https://web.archive.org/web/20190116195124/http://scifunam.fisica.unam.mx/mir/codes.html#haversine |date=2019-01-16 }}, [https://stackoverflow.com/a/4913653/1389451 Python], [https://github.com/kristianmandrup/haversine/blob/master/lib/haversine.rb Ruby], [http://www.movable-type.co.uk/scripts/LatLong.html JavaScript], [http://assemblysys.com/geographical-distance-calculation-in-php/ PHP] {{Webarchive|url=https://web.archive.org/web/20180812064108/http://assemblysys.com/geographical-distance-calculation-in-php/ |date=2018-08-12 }},[http://samoht.fr/informatique/distance-between-two-points-on-earth-surface-haversine-formula Matlab] {{Webarchive|url=https://web.archive.org/web/20200513140135/http://samoht.fr/informatique/distance-between-two-points-on-earth-surface-haversine-formula |date=2020-05-13 }}, [https://github.com/Lus71/lib_mysqludf_haversine MySQL]

Category:Spherical trigonometry

Category:Geodesy

Category:Distance