## Measuring Distance on the Earth

### Terminology

The words "geodesic" and "geodetic" are used in the following ways in geography.

1. When talking about coordinates, "geodetic longitude" and "geodetic latitude" are just fancy ways to say longitude and latitude. Hence, "geodetic coordinates" means the same thing as "geographic coordinates."
2. When talking about distance on the earth, "geodetic distance" and "geodesic distance" are the same thing: the shortest path along the ellipsoid of the earth at sea level between one point and another.

The OpenGIS Consortium refers to several kinds of distance.  The following is extracted from their abstract specification:

"The role of the reference system in distance calculations is important. Generally, there are at least three types of distances that may be defined between points (and therefore between geometric objects): map distance, geodetic distance, and terrain distance.

• Map distance is the distance between the points as defined by their position in a coordinate projection (such as on a map when scale is taken into account). Map distance is usually accurate for small areas where scale functions [are well-behaved].
• Geodetic distance is the length of the shortest curve between those two points along the surface of the earth model being used by the spatial reference system. Geodetic distance behaves well for wide areas of coverage, and takes the earth's curvature into account. It is especially handy for air and sea navigation, although care should be taken to distinguish between rhumb line (curves of constant geodetic bearing) and geodesic curve distance.
• Terrain distance will take into account the local vertical displacements (hypsography). Terrain distance can be based either on a geodetic distance or a map distance."

### Calculation of the Geodesic

• The simplest way to calculate geodesic distance is to find the angle between the two points, and multiply this by the circumference of the earth.  The formula is:
• angle = arccos(point1 * point2)
• distance = angle * pi * radius
• To be more accurate, you have to take into consideration the full ellipsoid, or even the geoid.  That's much more complicated than could be explained here (and way beyond my own grasp of cartography).  Fortunately there is functionality for this calculation in the free library PROJ.4, which is encapsulated and easier to use in the library vtdata.  For end-users, this calculation is exposed in the VTP Applications.
• There is more detail on the page PROJ.4: Geodesic Calculations

### Optimization

• See Efficient distances for discussion, with formulas, on how to optimize certain kinds of distance calculations