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HISTORY OF LONGITUDE

EARLY NAVIGATION

THE BRONZE AGE

ANCIENT FOUNDRIES

WEB LINKS ABOUT LONGITUDE

THE GEOCACHING GAME

BOOKS ON LONGITUDE

HOW GPS WORKS

GPS IN THE NEWS

GPS RECEIVERS

COORDINATES SOFTWARE

HELP FORMATING COORDINATES

MANY WAYS TO COMPUTE

COORDINATES

EARLY NAVIGATION

THE BRONZE AGE

ANCIENT FOUNDRIES

WEB LINKS ABOUT LONGITUDE

THE GEOCACHING GAME

BOOKS ON LONGITUDE

HOW GPS WORKS

GPS IN THE NEWS

GPS RECEIVERS

COORDINATES SOFTWARE

HELP FORMATING COORDINATES

MANY WAYS TO COMPUTE

COORDINATES

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.

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:

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

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:

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.**T***he 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:

Earth.

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.

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.)

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.)

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 |

CLASSROOM |