Problem Statement

The haversine formula is an equation important in navigation, giving great-circle distances between two points on a sphere from their longitudes and latitudes. It is a special case of a more general formula in spherical trigonometry, the law of haversines, relating the sides and angles of spherical "triangles".

Implement a great-circle distance function, or use a library function, to show the great-circle distance between:

Most of the examples below adopted Kaimbridge's recommended value of 6372.8 km for the earth radius. However, the derivation of this ellipsoidal quadratic mean radius is wrong (the averaging over azimuth is biased). When applying these examples in real applications, it is better to use the mean earth radius, 6371 km. This value is recommended by the International Union of Geodesy and Geophysics and it minimizes the RMS relative error between the great circle and geodesic distance.

Let's start with the solution:

define PI 
  -1 acos

define toRadians use $degree
  $degree PI * 180 /

define haversine use $lat1, $lon1, $lat2, $lon2
  6372.8 as $R
  # In kilometers
  $lat2 $lat1 - toRadians   as $dLat
  $lon2 $lon1 - toRadians   as $dLon
  $lat1 toRadians  as $lat1
  $lat2 toRadians  as $lat2
 
  $dLat 2 /  sin 
  $dLat 2 /  sin *
  $dLon 2 /  sin
  $dLon 2 /  sin *
  $lat1 cos * 
  $lat2 cos * +        as $a
  $a sqrt  asin  2 *   as $c
  $R $c *
}
 
-118.40 33.94 -86.67 36.12 haversine "haversine: %.15g\n" print