Simplest way to plot points randomly inside a circle

Question

I have a basic JSFiddle whereby I want to have random points plotted inside a circle.

But I do not know how to limit the points to be inside the circle.

This is what I currently have:

var ctx = canvas.getContext('2d'),
    count = 1000, // number of random  points
    cx = 150,
    cy = 150,
    radius = 148;

    ctx.beginPath();
    ctx.moveTo(cx, cy);
    ctx.arc(canvas.width/2, canvas.height/2, radius, 0, 2*Math.PI);
    ctx.closePath();
    ctx.fillStyle = '#00000';
    ctx.fill();

// create random points
    ctx.fillStyle = '#ffffff';

while(count) {
    // randomise x:y

    ctx.fillRect(x + canvas.width/2, y + canvas.height/2, 2, 2);
    count--;
}

How would i go about generating random (x,y) coordinates to plot random points inside the circle?

My current fiddle: http://jsfiddle.net/e8jqdxp3/

Answer 1

To plot points randomly in a circle, you can pick a random value from the radius squared, then square root it, pick a random angle, and convert the polar coordinate to rectangular. The square / square root step ensures that we get a uniform distribution (otherwise most points would be near the center of the circle).

So the formula to plot a random point in the circle is the following, where r' is a random value between 0 and r², and θ is a random value between 0 and 2π:

Screenshot of result:

Live Demo:

var canvas = document.getElementById("thecanvas");

var ctx = canvas.getContext('2d'),
    count = 1000, // number of random  points
    cx = 150,
    cy = 150,
    radius = 148;

ctx.fillStyle = '#CCCCCC';
ctx.fillRect(0, 0, canvas.width, canvas.height);

ctx.fillStyle = '#000000';

ctx.beginPath();
ctx.moveTo(cx, cy);
ctx.arc(canvas.width / 2, canvas.height / 2, radius, 0, 2 * Math.PI);
ctx.closePath();

ctx.fill();

// create random points
ctx.fillStyle = '#ffffff';

while (count) {
    var pt_angle = Math.random() * 2 * Math.PI;
    var pt_radius_sq = Math.random() * radius * radius;
    var pt_x = Math.sqrt(pt_radius_sq) * Math.cos(pt_angle);
    var pt_y = Math.sqrt(pt_radius_sq) * Math.sin(pt_angle);
    ctx.fillRect(pt_x + canvas.width / 2, pt_y + canvas.width / 2, 2, 2);
    count--;
}

<canvas id="thecanvas" width="400" height="400"></canvas>

JSFiddle Version: https://jsfiddle.net/qc735bqw/

Answer 2

Randomly pick dSquared (0..radius^2) and theta (0..2pi), then

x = sqrt(dSquared) cos(theta)
y = sqrt(dSquared) sin(theta)