# Documentation

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# distance

Distance between points on sphere or ellipsoid

## Syntax

`[arclen,az] = distance(lat1,lon1,lat2,lon2)[arclen,az] = distance(lat1,lon1,lat2,lon2,ellipsoid)[arclen,az] = distance(lat1,lon1,lat2,lon2,units)[arclen,az] = distance(lat1,lon1,lat2,lon2,ellipsoid,units)[arclen,az] = distance(track,...)[arclen,az] = distance(pt1,pt2)[arclen,az] = distance(pt1,pt2,ellipsoid)[arclen,az] = distance(pt1,pt,units)[arclen,az] = distance(pt1,pt2,ellipsoid,units)[arclen,az] = distance(track,pt1,...)`

## Description

`[arclen,az] = distance(lat1,lon1,lat2,lon2)` computes the lengths, `arclen`, of the great circle arcs connecting pairs of points on the surface of a sphere. In each case, the shorter (minor) arc is assumed. The function can also compute the azimuths, `az`, of the second point in each pair with respect to the first (that is, the angle at which the arc crosses the meridian containing the first point). The input latitudes and longitudes, `lat1`, `lon1`, `lat2`, `lon2`, can be scalars or arrays of equal size and must be expressed in degrees. `arclen` is expressed in degrees of arc and will have the same size as the input arrays. `az` is measured clockwise from north, in units of degrees. When given a combination of scalar and array inputs, the scalar inputs are automatically expanded to match the size of the arrays.

`[arclen,az] = distance(lat1,lon1,lat2,lon2,ellipsoid)` computes geodesic arc length and azimuth assuming that the points lie on the reference ellipsoid defined by the input `ellipsoid`. `ellipsoid` is a `referenceSphere`, `referenceEllipsoid`, or `oblateSpheroid` object, or a vector of the form `[semimajor_axis eccentricity]`. The output, `arclen`, is expressed in the same length units as the semimajor axis of the ellipsoid.

`[arclen,az] = distance(lat1,lon1,lat2,lon2,units)` where `units` defines the angle unit of the outputs `arclen` and `az`, and the input latitude-longitude coordinates. `units` may equal `'degrees'` (the default value) or `'radians'`.

`[arclen,az] = distance(lat1,lon1,lat2,lon2,ellipsoid,units)` where `units` specifies the units of the latitude-longitude coordinates, but the output range has the same units as the semimajor axis of the ellipsoid.

`[arclen,az] = distance(track,...)` where `track` specifies either a great circle/geodesic or a rhumb line arc. If `track` equals `'gc'` (the default value), then great circle distances are computed on a sphere and geodesic distances are computed on an ellipsoid. If `track` equals `'rh'`, then rhumb line distances are computed on either a sphere or ellipsoid.

`[arclen,az] = distance(pt1,pt2)` accepts N-by-2 coordinate arrays `pt1` and `pt2` such that `pt1 = [lat1 lon1]` and ```pt2 = [lat2 lon2]```, where `lat1`, `lon1`, `lat2`, and `lon2` are column vectors. It is equivalent to ```arclen = distance(pt1(:,1),pt1(:,2),pt2(:,1),pt2(:,2))```.

`[arclen,az] = distance(pt1,pt2,ellipsoid)`,

`[arclen,az] = distance(pt1,pt,units)`,

`[arclen,az] = distance(pt1,pt2,ellipsoid,units)`, and

`[arclen,az] = distance(track,pt1,...)` are all valid calling forms.

## Examples

Using `pt1,pt2` notation, find the distance from Norfolk, Virginia (37ºN, 76ºW), to Cape St. Vincent, Portugal (37ºN, 9ºW), just outside the Straits of Gibraltar. The distance between these two points depends upon the `track` value selected.

```arclen = distance('gc',[37,-76],[37,-9]) arclen = 52.3094 arclen = distance('rh',[37,-76],[37,-9]) arclen = 53.5086```

The difference between these two tracks is 1.1992 degrees, or about 72 nautical miles. This represents about 2% of the total trip distance. The trade-off is that at the cost of those 72 miles, the entire trip can be made on a rhumb line with a fixed course of 90º, due east, while in order to follow the shorter great circle path, the course must be changed continuously.

On a meridian and on the Equator, great circles and rhumb lines coincide, so the distances are the same. For example,

```% Great circle distance arclen = distance(37,-76,67,-76) arclen = 30.0000 % Rhumb line distance arclen = distance('rh',37,-76,67,-76) arclen = 30.0000```

The distances are the same, 30º, or about 1800 nautical miles. (There are about 60 nautical miles in a degree of arc length.)

## Alternatives

Distance between two points can be calculated in two ways. For great circles (on the sphere) and geodesics (on the ellipsoid), the distance is the shortest surface distance between two points. For rhumb lines, the distance is measured along the rhumb line passing through the two points, which is not, in general, the shortest surface distance between them.

When you need to compute both distance and azimuth for the same point pair(s), it is more efficient to do so with a single call to `distance`. That is, use

```[arclen az] = distance(...); ```
rather than the slower
```arclen = distance(...) az = azimuth(...)```

To express the output `arclen` as an arc length in either degrees or radians, omit the `ellipsoid` argument. This is possible only on a sphere. If `ellipsoid` is supplied, `arclen` is a distance expressed in the same units as the semimajor axis of the ellipsoid. Specify `ellipsoid` as ```[R 0]``` to compute `arclen` as a distance on a sphere of radius `R`, with `arclen` having the same units as `R`.

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### Algorithms

Distance calculations for geodesics degrade slowly with increasing distance and may break down for points that are nearly antipodal, as well as when both points are very close to the Equator. In addition, for calculations on an ellipsoid, there is a small but finite input space, consisting of pairs of locations in which both the points are nearly antipodal and both points fall close to (but not precisely on) the Equator. In this case, a warning is issued and both `arclen` and `az` are set to NaN for the "problem pairs."