# Arc and distance between two points on Earth surface

The following C++ snippet computes the arc (in radians) and the distance (in meters) between two positions on Earth using the law of haversines.

It assumes a `Position`

class exists, with two public members for the longitude and the latitude (resp. "lon" and "lat").

/// @brief The usual PI/180 constant static const double DEG_TO_RAD = 0.017453292519943295769236907684886; /// @brief Earth's quatratic mean radius for WGS-84 static const double EARTH_RADIUS_IN_METERS = 6372797.560856; /** @brief Computes the arc, in radian, between two WGS-84 positions. * * The result is equal to <code>Distance(from,to)/EARTH_RADIUS_IN_METERS</code> * <code>= 2*asin(sqrt(h(d/EARTH_RADIUS_IN_METERS )))</code> * * where:<ul> * <li>d is the distance in meters between 'from' and 'to' positions.</li> * <li>h is the haversine function: <code>h(x)=sin²(x/2)</code></li> * </ul> * * The haversine formula gives: * <code>h(d/R) = h(from.lat-to.lat)+h(from.lon-to.lon)+cos(from.lat)*cos(to.lat)</code> * * @sa http://en.wikipedia.org/wiki/Law_of_haversines */ double ArcInRadians(const Position& from, const Position& to) { double latitudeArc = (from.lat - to.lat) * DEG_TO_RAD; double longitudeArc = (from.lon - to.lon) * DEG_TO_RAD; double latitudeH = sin(latitudeArc * 0.5); latitudeH *= latitudeH; double lontitudeH = sin(longitudeArc * 0.5); lontitudeH *= lontitudeH; double tmp = cos(from.lat*DEG_TO_RAD) * cos(to.lat*DEG_TO_RAD); return 2.0 * asin(sqrt(latitudeH + tmp*lontitudeH)); } /** @brief Computes the distance, in meters, between two WGS-84 positions. * * The result is equal to <code>EARTH_RADIUS_IN_METERS*ArcInRadians(from,to)</code> * * @sa ArcInRadians */ double DistanceInMeters(const Position& from, const Position& to) { return EARTH_RADIUS_IN_METERS*ArcInRadians(from, to); }

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