Smart up this.

brute-force.js

calculate_random_point: function(existing_points, min_angle, max_angle, plot_radius, point_radius) {
    var angle = this.random_angle(min_angle, max_angle);
    var point_radius = Math.random() * (plot_radius - (point_radius / 2));
    var point = {
        x: (Math.sin(angle) * point_radius),
        y: (Math.cos(angle) * point_radius)
    };
    if (_.any(existing_points, function(existing_point) {
        var distance = Math.sqrt(Math.pow(existing_point.x - point.x,2) + Math.pow(existing_point.y - point.y,2));
            return distance < point_radius;
        })) {
            log("Bummer there is a point to close to that one (" + point.x +", " + point.y +"). Trying again.");
            return this.calculate_random_point(existing_points, min_angle, max_angle, plot_radius, point_radius);
        } else {
            return point;
        }
    }
},
random_angle: function(min_angle, max_angle) {
    return (Math.random() * (max_angle - min_angle) + min_angle) * (Math.PI/180);
}

Reading your gist-comment i interpret it as trying to make a circle with many dots plotted on its circumference. It is allso a requirement that they do not overlap. If this is the case i would not use any Math.random...
But there is a chance i have not understood your req-spec at all :):):):)
I get association of an old "dial-disc telephone" or google friend circles..

function GetDotPlotPositionsOnCircumference(centerx,centery, cradius, dotradius, startangle, endangle, noOfDotsNeeded )
{
  var nicePositions = [];
  var minAngleBetweenDots =360 * (2*pi*cRadius) / (2*dotradius);
  var deltaAngle = (endAngle - startangle)/noOfDotsNeeded;

  if (deltaAngle<minAngleBetweenDots) {
    console.log("To many dots for such a smal arc.");   
  } else{

     for (int i = 0; i<noOfPointsNeeded){
          var point = {
            x: centerx+(Math.sin(i*minAngleBetweenDots+startangle) * cradius),
            y: centery+(Math.cos(i*minAndleBetweenDots+startangle) * cradius)
              };
          nicePositions.push(points);
      }
    }
   return nicePositions;
}