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.
The first table of haversines in English was published by James Andrew in 1805,[1] but Florian Cajori credits an earlier use by José de Mendoza y Ríos in 1801.[2][3] The term haversine was coined in 1835 by James Inman.[4][5]
These names follow from the fact that they are customarily written in terms of the haversine function, given by haversin(θ) = sin2(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 sin2 function.
Formulation[edit]
Let the central angle Θ between any two points on a sphere be:
-
Θ
=d
r{displaystyle Theta ={frac {d}{r}}}
where:
The haversine formula hav of Θ is given by:
-
hav
(
Θ
)=
hav
(
φ
2
−
φ
1
)
+
cos
(
φ
1
)
cos
(
φ
2
)
hav
(
λ
2
−
λ
1
)
{displaystyle 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
- φ1, φ2: latitude of point 1 and latitude of point 2,
- λ1, λ2: longitude of point 1 and longitude of point 2.
Finally, the haversine function (half a versine) of an angle θ (applied above to the differences in latitude and longitude) is:
-
hav
(
θ
)
=sin
2
(
θ
2)
=
1
−
cos
(
θ
)2
{displaystyle operatorname {hav} (theta )=sin ^{2}left({frac {theta }{2}}right)={frac {1-cos(theta )}{2}}}
To solve for the distance d, apply the archaversine (inverse haversine) to the central angle Θ or use the arcsine (inverse sine) function:
-
d
=
r
archav
(
h
)
=
2
r
arcsin
(
h
)
{displaystyle d=roperatorname {archav} (h)=2rarcsin left({sqrt {h}}right)}
where h = hav(Θ), or more explicitly:
-
d
=
2
r
arcsin
(
hav
(φ
2
−
φ
1
)
+
cos
(φ
1
)
cos
(φ
2
)
hav
(λ
2
−
λ
1
)
)
=
2
r
arcsin
(
sin
2
(
φ
2
−
φ
1
2
)
+
cos
(φ
1
)
cos
(φ
2
)
sin
2
(
λ
2
−
λ
1
2
)
)
{displaystyle {begin{aligned}d&=2rarcsin left({sqrt {operatorname {hav} (varphi _{2}-varphi _{1})+cos(varphi _{1})cos(varphi _{2})operatorname {hav} (lambda _{2}-lambda _{1})}}right)\&=2rarcsin left({sqrt {sin ^{2}left({frac {varphi _{2}-varphi _{1}}{2}}right)+cos(varphi _{1})cos(varphi _{2})sin ^{2}left({frac {lambda _{2}-lambda _{1}}{2}}right)}}right)end{aligned}}}
When using these formulae, one must ensure that h does not exceed 1 due to a floating point error (d is only real for h from 0 to 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 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 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) you 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[edit]
Spherical triangle solved by 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
{displaystyle u}
,
v
{displaystyle v}
, and
w
{displaystyle w}
on the sphere. If the lengths of these three sides are
a
{displaystyle a}
(from
u
{displaystyle u}
to
v
{displaystyle v}
),
b
{displaystyle b}
(from
u
{displaystyle u}
to
w
{displaystyle w}
), and
c
{displaystyle c}
(from
v
{displaystyle v}
to
w
{displaystyle w}
), and the angle of the corner opposite
c
{displaystyle c}
is
C
{displaystyle C}
, then the law of haversines states:
-
hav
(
c
)
=
hav
(
a
−
b
)
+
sin
(
a
)
sin
(
b
)
hav
(
C
)
.{displaystyle operatorname {hav} (c)=operatorname {hav} (a-b)+sin(a)sin(b)operatorname {hav} (C).}
[9]
Since this is a unit sphere, the lengths
a
{displaystyle a}
,
b
{displaystyle b}
, and
c
{displaystyle 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
{displaystyle 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
{displaystyle u}
is the north pole, while
v
{displaystyle v}
and
w
{displaystyle w}
are the two points whose separation
d
{displaystyle d}
is to be determined. In that case,
a
{displaystyle a}
and
b
{displaystyle b}
are
π
2
−
φ
1
,
2
{displaystyle {frac {pi }{2}}-varphi _{1,2}}
(i.e., co-latitudes),
C
{displaystyle C}
is the longitude separation
Δ
λ
{displaystyle Delta lambda }
, and
c
{displaystyle c}
is the desired
d
R
{displaystyle {frac {d}{R}}}
. Noting that
sin
(
π
2
−
φ
)
=
cos
(
φ
)
{displaystyle sin left({frac {pi }{2}}-varphi right)=cos(varphi )}
, 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
)
.{displaystyle 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
{displaystyle c}
when
c
{displaystyle c}
is small. Instead, we substitute the identity that
cos
(
θ
)
=
1
−
2
hav
(
θ
)
{displaystyle cos(theta )=1-2operatorname {hav} (theta )}
, and also employ the addition identity
cos
(
a
−
b
)
=
cos
(
a
)
cos
(
b
)
+
sin
(
a
)
sin
(
b
)
{displaystyle cos(a-b)=cos(a)cos(b)+sin(a)sin(b)}
, to obtain the law of haversines, above.
See also[edit]
References[edit]
- ^ 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 aplication 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. 2 (2 (3rd corrected printing of 1929 issue) ed.). Chicago, USA: 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 2nd edition by Cosimo, Inc., New York, USA, 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: [1].)
- ^ “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).
- ^ 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 (3 ed.). Mineola, New York, USA: Dover Publications, Inc. pp. 892–893. ISBN 978-0-486-41147-7.
Further reading[edit]
- U. S. Census Bureau Geographic Information Systems FAQ, (content has been moved to 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).
- Deriving the haversine formula, Ask Dr. Math (Apr. 20–21, 1999).
- Romuald Ireneus ‘Scibor-Marchocki, Spherical trigonometry, Elementary-Geometry Trigonometry web page (1997).
- 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).