How to generate random LAT & LNG inside an area, given the center and the radius

Question

In my project my users can choose to be put in a random position inside a given, circular area. I have the latitude and longitude of the center and the radius: how can I calculate the latitude and longitude of a random point inside the given area? (I use PHP but examples in any language will fit anyway)

Answer 1

You need two randomly generated numbers.

Thinking about this using rectangular (Cartesian) (x,y) coordinates is somewhat unnatural for the problem space. Given a radius, it's somewhat difficult to think about how to directly compute an (Δx,Δy) delta that falls within the circle defined by the center and radius.

Better to use polar coordinates to analyze the problem - in which the dimensions are (r1, Θ). Compute one random distance, bounded by the radius. Compute a random angle, from 0 to 360 degrees. Then convert the (r,Θ) to Cartesian (Δx,Δy), where the Cartesian quantities are simply offsets from your circle center, using the simple trigonometry relations.

  Δx = r * cos(Θ)
  Δy = r * sin(Θ)

Then your new point is simply

  xnew = x + Δx
  ynew = y + Δy

This works for small r, in which case the geometry of the earth can be approximated by Euclidean (flat plane) geometry.

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As r becomes larger, the curvature of the earth means that the Euclidean approximation does not match the reality of the situation. In that case you will need to use the formulas for geodesic distance, which take into account the 3d curvature of the earth. This begins to make sense, let's say, above distances of 100 km. Of course it depends on the degree of accuracy you need, but I'm assuming you have quite a bit of wiggle room.

In 3d-geometry, you once again need to compute 2 quantities - the angle and the distance. The distance is again bound by your r radius, except in this case the distance is measured on the surface of the earth, and is known as a "great circle distance". Randomly generate a number less than or equal to your r, for the first quantity.

The great-circle geometry relation

 d =  R Δσ

...states that d, a great-circle distance, is proportional to the radius of the sphere and the central angle subtended by two points on the surface of the sphere. "central angle" refers to an angle described by three points, with the center of the sphere at the vertex, and the other two points on the surface of the sphere.

In your problem, this d must be a random quantity bound by your original 'r'. Calculating a d then gives you the central angle, in other words Δσ , since the R for the earth is known (about 6371.01 km).

That gives you the absolute (random) distance along the great circle, away from your original lat/long. Now you need the direction, quantified by an angle, describing the N/S/E/W direction of travel from your original point. Again, use a 0-360 degree random number, where zero represents due east, if you like.

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The change in latitude can be calculated by d sin(Θ) , the change in longitude by d cos(Θ). That gives the great-circle distance in the same dimensions as r (presumably km), but you want lat/long degrees, so you'll need to convert. To get from latitudinal distance to degrees is easy: it's about 111.32 km per degree regardless of latitude. The conversion from longitudinal distance to longitudinal degrees is more complicated, because the longitudinal lines are closer to each other nearer the poles. So you need to use some more complex formulae to compute the change in longitude corresponding to the selected d (great distance) and angle. Remember you may need to hop over the +/- 180° barrier. (The designers of the F22 Raptor warplane forgot this, and their airplanes nearly crashed when attempting to cross the 180th meridian.)

Because of the error that may accumulate in successive approximations, you will want to check that the new point fits your constraints. Use the formula

Δσ  = arccos( cos(Δlat) - cos(lat1)*cos(lat2)*(1 - cos(Δlong) ) .

where Δlat is the change in latitude, etc.

This gives you Δσ , the central angle between the new and old lat/long points. Verify that the central angle you've calcuated here is the same as the central angle you randomly selected previously. In other words verify that the computed d (great-circle-distance) between the calculated points is the same as the great circle distance you randomly selected. If the computed d varies from your selected d, you can use numerical approximation to improve the accuracy, altering the latitude or longitude slightly until it meets your criterion.

Answer 2

This can simply be done by calculating a random bearing (between 0 and 2*pi) and a random distance between 0 and your desired maximum radius. Then compute the new (random) lat/lon using the random bearing/range from your given lat/lon center point. See the section 'Destination point given distance and bearing from start point' at this web site: http://www.movable-type.co.uk/scripts/latlong.html

Note: the formula given expects all angles as radians (including lat/lon). The resulting lat/lon with be in radians also so you will need to convert to degrees.