Vincenty's formulae are two related
iterative method
In computational mathematics, an iterative method is a Algorithm, mathematical procedure that uses an initial value to generate a sequence of improving approximate solutions for a class of problems, in which the ''n''-th approximation is derived fr ...
s used in
geodesy
Geodesy ( ) is the Earth science of accurately measuring and understanding Earth's figure (geometric shape and size), orientation in space, and gravity. The field also incorporates studies of how these properties change over time and equivale ...
to calculate the distance between two points on the surface of a spheroid, developed by
Thaddeus Vincenty (1975a). They are based on the assumption that the
figure of the Earth
Figure of the Earth is a Jargon, term of art in geodesy that refers to the size and shape used to model Earth. The size and shape it refers to depend on context, including the precision needed for the model. A Spherical Earth, sphere is a well-k ...
is an
oblate spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface obtained by rotating an ellipse about one of its principal axes; in other words, an ellipsoid with two equal semi-diameters. A spheroid has circ ...
, and hence are more accurate than methods that assume a
spherical
A sphere () is a geometrical object that is a three-dimensional analogue to a two-dimensional circle. A sphere is the set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the ce ...
Earth, such as
great-circle distance
The great-circle distance, orthodromic distance, or spherical distance is the distance along a great circle.
It is the shortest distance between two points on the surface of a sphere, measured along the surface of the sphere (as opposed to a ...
.
The first (direct) method computes the location of a point that is a given distance and
azimuth
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematicall ...
(direction) from another point. The second (inverse) method computes the
geographical distance
Geographical distance or geodetic distance is the distance measured along the surface of the earth. The formulae in this article calculate distances between points which are defined by geographical coordinates in terms of latitude and longitude. ...
and
azimuth
An azimuth (; from ar, اَلسُّمُوت, as-sumūt, the directions) is an angular measurement in a spherical coordinate system. More specifically, it is the horizontal angle from a cardinal direction, most commonly north.
Mathematicall ...
between two given points. They have been widely used in geodesy because they are accurate to within 0.5 mm (0.020in) on the
Earth ellipsoid
An Earth ellipsoid or Earth spheroid is a mathematical figure approximating the Earth's form, used as a reference frame for computations in geodesy, astronomy, and the geosciences. Various different ellipsoids have been used as approximations ...
.
Background
Vincenty's goal was to express existing algorithms for
geodesics on an ellipsoid
The study of geodesics on an ellipsoid arose in connection with geodesy specifically with the solution of triangulation networks. The figure of the Earth is well approximated by an ''oblate ellipsoid'', a slightly flattened sphere. A ''geodes ...
in a form that minimized the program length (Vincenty 1975a). His unpublished report (1975b) mentions the use of a
Wang
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Names
* Wang (surname) (王), a common Chinese surname
* Wāng (汪), a less common Chinese surname
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* A title in Korean nobility
* A title in Mongolian nobility
Places
* Wang River in Thailand ...
720 desk calculator, which had only a few kilobytes of memory. To obtain good accuracy for long lines, the solution uses the classical solution of Legendre (1806), Bessel (1825), and Helmert (1880) based on the auxiliary sphere. Vincenty relied on formulation of this method given by Rainsford, 1955. Legendre showed that an ellipsoidal geodesic can be exactly mapped to a great circle on the auxiliary sphere by mapping the geographic latitude to reduced latitude and setting the azimuth of the great circle equal to that of the geodesic. The longitude on the ellipsoid and the distance along the geodesic are then given in terms of the longitude on the sphere and the arc length along the great circle by simple integrals. Bessel and Helmert gave rapidly converging series for these integrals, which allow the geodesic to be computed with arbitrary accuracy.
In order to minimize the program size, Vincenty took these series, re-expanded them using the first term of each series as the small parameter, and truncated them to
. This resulted in compact expressions for the longitude and distance integrals. The expressions were put in
Horner
Horner is an English and German surname that derives from the Middle English word for the occupation ''horner'', meaning horn-worker or horn-maker, or even horn-blower.
People
*Alison Horner (born 1966), British businesswoman
* Arthur Horner (dis ...
(or ''nested'') form, since this allows polynomials to be evaluated using only a single temporary register. Finally, simple iterative techniques were used to solve the implicit equations in the direct and inverse methods; even though these are slow (and in the case of the inverse method it sometimes does not converge), they result in the least increase in code size.
Notation
Define the following notation:
Inverse problem
Given the coordinates of the two points (''Φ''
1, ''L''
1) and (''Φ''
2, ''L''
2), the inverse problem finds the azimuths ''α''
1, ''α''
2 and the ellipsoidal distance ''s''.
Calculate ''U''
1, ''U''
2 and ''L'', and set initial value of ''λ'' = ''L''. Then iteratively evaluate the following equations until ''λ'' converges:
:
:
:
[''σ'' is not evaluated directly from sin ''σ'' or cos ''σ'' to preserve numerical accuracy near the poles and equator]
:
[If sin ''σ = 0'' the value of sin ''α'' is indeterminate. It represents an end point coincident with, or diametrically opposed to, the start point.]
:
[Where the start and end point are on the equator, and the value of is not used. The limiting value is .]
: