Algorithm: Calculate pseudo-random point inside an ellipse

Question

For a simple particle system I'm making, I need to, given an ellipse with width and height, calculate a random point X, Y which lies in that ellipse.

Now I'm not the best at maths, so I wanted to ask here if anybody could point me in the right direction.

Maybe the right way is to choose a random float in the range of the width, take it for X and from it calculate the Y value?

Answer 1

1. Generate a random point inside a circle of radius 1. This can be done by taking a random angle phi in the interval [0, 2*pi) and a random value rho in the interval [0, 1) and compute

   x = sqrt(rho) * cos(phi)
   y = sqrt(rho) * sin(phi)
   

   The square root in the formula ensures a uniform distribution inside the circle.

2. Scale x and y to the dimensions of the ellipse

   x = x * width/2.0
   y = y * height/2.0