Random points all over inside of a circle

Question

I'm using below code to randomly generate some x,y s that are located inside of a circle:

// r is distance from the center and a is angle
// R is radius of circle
// M is center

r = R * Math.random();
a = 2 * Math.PI * Math.random();

x = Math.round(r * Math.cos(a) + M);
y = Math.round(r * Math.sin(a) + M);

the problem is while moving closer and closer to the center of the circle, the chance to get x,y in that location gets more and more.

But What I'm looking for is just to have completely random x,y all over in the circle. How can I achieve this?

Answer 1

Due to Jacobian, you have to take square root

r = R *  Math.sqrt(Math.random());

Answer 2

To evenly distribute points within the circle, just pick random points in a square with side-length equal to 2R, and then discard any point that falls outside of the circle.

This can be done without any trigonometric or transcendental operations:

let x, y;
do {
    x = 2 * Math.random() - 1.0;  // range [-1, +1)
    y = 2 * Math.random() - 1.0;
} while ((x * x + y * y) >= 1);   // check unit circle

// scale and translate the points
x = x * R + Mx;
y = y * R + My;

In each pass of the loop about 21.5% of points will be discarded but this should still end up faster than using sin and cos.