Picking random points in elliptic arc or sector

This was the result of random flashkit thread encounter and has no much real life value, probably. The closest thing I found to what that guy was asking for was this 2009 post explaining the trick to pick random point in the circle, but it is remarkably easy to extend it to the ellipse, or even arbitrary elliptic arc or sector. It is interesting in that you pick correct point on 1st try versus, for example, 4/π ≈ 1.27-th try if you simply select random point in ellipse bounding box and reject outliers – and some extra Math.xxx calls pay for that.