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Relationship Between Points on Sphere

When using spherical coordinates, distances are expressed as angles, not lengths. As there is an infinity of arcs that can connect two points on a sphere or spheroid, by convention the shortest one (the great circle distance) is used to measure how close two points are. As is explained in Distances on the Sphere, you can convert angular distance on a sphere to linear distance. This is different from working on an ellipsoid, where one can only speak of linear distances between points, and to compute them one must specify which reference ellipsoid to use.

In spherical or geodetic coordinates, a position is a latitude taken together with a longitude, e.g., (lat,lon), which defines the horizontal coordinates of a point on the surface of a planet. When we consider two points, e.g.,(lat1,lon1) and (lat2,lon2), there are several ways in which their 2–D spatial relationships are typically quantified:

  • The azimuth (also called heading) to take to get from (lat1,lon1) to (lat2,lon2)

  • The back azimuth (also called heading) from (lat2,lon2) to (lat1,lon1)

  • The spherical distance separating (lat1,lon1) from (lat2,lon2)

  • The linear distance (range) separating (lat1,lon1) from (lat2,lon2)

The first three are angular quantities, while the last is a length. Mapping Toolbox™ functions exist for computing these quantities. For additional examples, see Navigation.

There is no single default unit of distance measurement in the toolbox. Navigation functions use nautical miles as a default and the distance function uses degrees of arc length. For many functions, the default unit for distances and positions is degrees, but you need to verify the default assumptions before using any of these functions.


When distances are given in terms of angular units (degrees or radians), be careful to remember that these are specified in terms of arc length. While a degree of latitude always subtends one degree of arc length, this is only true for degrees of longitude along the equator.