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 ''i''-th approximation (called an " ...
s used in
geodesy
Geodesy or geodetics is the science of measuring and representing the Figure of the Earth, geometry, Gravity of Earth, gravity, and Earth's rotation, spatial orientation of the Earth in Relative change, temporally varying Three-dimensional spac ...
to calculate the
distance
Distance is a numerical or occasionally qualitative measurement of how far apart objects, points, people, or ideas are. In physics or everyday usage, distance may refer to a physical length or an estimation based on other criteria (e.g. "two co ...
between two
points on the surface of a
spheroid
A spheroid, also known as an ellipsoid of revolution or rotational ellipsoid, is a quadric surface (mathematics), surface obtained by Surface of revolution, rotating an ellipse about one of its principal axes; in other words, an ellipsoid with t ...
, developed by
Thaddeus Vincenty (1975a). They are based on the assumption that the
figure of the Earth
In geodesy, the figure of the Earth is the size and shape used to model planet Earth. The kind of figure depends on application, including the precision needed for the model. A spherical Earth is a well-known historical approximation that is ...
is an oblate spheroid, and hence are more accurate than methods that assume a
spherical Earth, such as
great-circle distance
The great-circle distance, orthodromic distance, or spherical distance is the distance between two points on a sphere, measured along the great-circle arc between them. This arc is the shortest path between the two points on the surface of the ...
.
The first (direct) method computes the location of a point that is a given distance and
azimuth
An azimuth (; from ) is the horizontal angle from a cardinal direction, most commonly north, in a local or observer-centric spherical coordinate system.
Mathematically, the relative position vector from an observer ( origin) to a point ...
(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, or the shortest arch length.
The formulae in this article calculate distances between points which are defined by geographical coordinates in t ...
and
azimuth
An azimuth (; from ) is the horizontal angle from a cardinal direction, most commonly north, in a local or observer-centric spherical coordinate system.
Mathematically, the relative position vector from an observer ( origin) to a point ...
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 approximation ...
.
Background
Vincenty's goal was to express existing algorithms for
geodesics on an ellipsoid in a form that minimized the program length (Vincenty 1975a). His unpublished report (1975b) mentions the use of a
Wang
Wang may refer to:
Names
* Wang (surname)
Wang () is the pinyin romanization of Chinese, romanization of the common Chinese surname (''Wáng''). It has a mixture of various origin with uncertain lineage of family history, however it is c ...
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 (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 .]
: