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The Haversine Formula
The Haversine Formula
For two points on a sphere (of radius R) with latitudes φ1 and φ2,
latitude separation Δφ = φ1 − φ2, and longitude separation Δλ, where
angles are in radians, the distance d between the two points (along a
great circle of the sphere; see spherical distance) is related to their
locations by the formula above.
Let h denote haversin(d/R), given from above. One can then solve for d
either by simply applying the inverse haversine (if available) or by using
the arcsine (inverse sine) function:
In the era before the digital calculator, the use of detailed
printed tables for the haversine/inverse-haversine and its
logarithm (to aid multiplications) saved navigators from
squaring sines, computing square roots, etc., a process both
arduous and likely to exacerbate small errors (see also
versine).

When using these formulae, care must be taken to 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. However, because d is
then large (approaching πR, half the circumference) a small
error is often not a major concern in this unusual case
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 of the sphere).n 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 - φ1,2
(i.e., 90° − latitude), C is the longitude separation Δλ, 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:
This formula is only an approximation when applied to the
Earth.  The Earth is not a perfect sphere: The radius R
varies from 6356.78 km at the poles to 6378.14 km at the
equator. There are small corrections, typically on the order
of 0.1% (assuming the geometric mean R = 6367.45 km is
used everywhere), because of this slight ellipticity of the
planet. A more accurate method, which takes into account
the Earth's ellipticity, is given by Vincenty's formulae.

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:
As described below, a similar formula can also be written in
terms of cosines (sometimes called the spherical law of
cosines, not to be confused with the law of cosines for plane
geometry) instead of haversines, but for the common case of
small distances/angles a small error in the input to the
arccos function leads to a large error in the final output. This
makes the formula unsuitable for general use.
(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.)
Navigators used logs to get round the difficulties
they always had doing long-multiplication and
long-division. The problem was that logarithms
couldn't be used with negative numbers, the log of a
negative number being a meaningless concept. As
ordinary trig functions ranged over positive and
negative values, something had to be done.

So a new trig function, the "versine" was invented,
which never went negative, but varied between
zero and +2. The haversine, as you might guess, is
simply half the versine, so it varies, more
conveniently, between zero and +1. And then all the
trig formulae that navigators used were bent and
twisted into forms that used one of these new
functions, and nothing that required a log ever
took a negative value.
Versine vs. Haversine
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