How do I generate a random latitude and longitude within the radius of another point?

Question

In Ruby if I have a particular latitude and longitude, how can I generate a relatively random sample of points within a given radius or near another point?

{ latitude: 37.7905576, longitude: -122.3989885 }

Answer 1

Theory

The circumference of the earth at equator = 40,076 km. The equator is divided into 360 degrees of longitude, so each degree at the equator represents approximately 111.32 km. Moving away from the equator towards a pole this distance decreases to zero at the pole.

1 degree aproximates to 111.32 km at equator.
96.41km at 30 degrees N/S
78.71 km at 45 degrees N/S
55.66 km at 60 degrees N/S
28.82 km at 75 degrees N/S

Therefor just adding or subtracting a random distance from the coordinates will not produce a "circle" . To avoid this we can generate a series of points and eliminating those outside the radius by using either Haversine,Spherical Law Of Cosines or Equirectangular projection.

As the latter uses only the common trig functions cos() I will use it to illustrate. Using php as I don't have Ruby

Application

function Equirectangular($lat1,$lng1,$lat2,$lng2){
$x = deg2rad($lng2-$lng1) * cos(deg2rad($lat1+$lat2)/2);
$y = deg2rad($lat1-$lat2);
$R = 6372.8; // gives d in km
$distance = sqrt($x*$x + $y*$y) * $R;
return $distance;
}

$centerLat = 37.7905576;
$centerLng = -122.3989885; 


for ($i = 0; $i <= 200; $i++) {
    $lat = rand(0,40000)/2;//Random between 0 & 2 =radius +- 2 degrees 222.62 km
    $lng = rand(0,40000)/2;
    $lat2 = $centerLat+ ($lat/10000);
    $lng2 = $centerLng+ ($lng/10000);
    echo $lat2."  ".$lng2."";
    if (Equirectangular($centerLat,$centerLng,$lat2,$lng2)< 200){//200 km 
    echo " In";
    }else{
    echo " Out";
    }
    echo "<br>";  
}