Mapping Toolbox

Reckoning -- the Forward Problem

A common problem in geographic applications is the determination of a destination given a starting point, an initial azimuth, and a distance. In the Mapping Toolbox, this process is called reckoning. A new position can be reckoned in a great circle or a rhumb line sense (great circle or rhumb line track).

As an example, an airplane takes off from La Guardia Airport in New York (40.75ºN, 73.9ºW) and follows a northwestern rhumb line flight path at 200 knots (nautical miles per hour). Where would it be after 1 hour?

• [rhlat,rhlong] = reckon('rh',40.75,-73.9,nm2deg(200),315)
  rhlat =
      43.1054
  rhlong =
      -77.0665
  

Notice that the distance, 200 nautical miles, must be converted to degrees of arc length with the nm2deg conversion function to match the latitude and longitude inputs. If the airplane had a flight computer that allowed it to follow an exact great circle path, what would the aircraft's new location be?

• [gclat,gclong] = reckon('gc',40.75,-73.9,nm2deg(200),315)
  gclat =
      43.0615
  gclong =
      -77.1238
  

Notice also that for short distances at these latitudes, the result hardly differs between great circle and rhumb line. The two destination points are less than 4 nautical miles apart. Incidentally, after 1 hour, the airplane would be just north of New York's Finger Lakes.

Calculating Tracks - Great Circles and Rhumb Lines

You can generate vector data corresponding to points along great circle or rhumb line tracks using track1 and track2. If you have a point on the track and an azimuth at that point, use track1. If you have two points on the track, use track2. For example, to get the great circle path starting at (31ºS, 90ºE) with an azimuth of 45º with a length of 12º, use track1:

• [latgc,longc] = track1('gc',-31,90,45,12);
  

For the great circle from (31ºS, 90ºE) to (23ºS, 110ºE), use track2:

• [latgc,longc] = track2('gc',-31,90,-23,110);
  
  
  

The track1 function also allows you to specify range endpoints. For example, if you want points along a rhumb line starting 5º away from the initial point and ending 13º away, at an azimuth of 55º, simply specify the range limits:

• [latrh,lonrh] = track1('rh',-31,90,55,[5 13]);
  
  
  

When no range is provided for track1, the returned points represent a complete track. For great circles, a complete track is 360º, encircling the planet and returning to the initial point. For rhumb lines, the complete track terminates at the poles, unless the azimuth is 90º or 270º, in which case the complete track is a parallel that returns to the initial point.

For calculated tracks, 100 points are returned unless otherwise specified. You can calculate several tracks at one time by providing vector inputs. For more information, see the track1 and track2 functions in the online Mapping Toolbox reference documentation. More vector path calculations are described later in Navigation.

Angles and Directions on the Sphere and Spheroid

Distance, Azimuth, and Back-azimuth (the Inverse Problem)