Generate a random vector inside a rectangle but not a circle?

Question

I have a rectangle, and a circle inside that rectangle (that sits around the center of the rectangle). I want to generate a random 2-component vector that falls inside the rectangle, but not the circle. How can I do it?

Edit: I'd prefer a method that i can use to generate a vector that meets these constraints without brute-forcing it.

Answer 1

Vector = Rectangle.RandomVector();

while (Circle.Contains(Vector)) {
    Vector = Rectangle.RandomVector();
}

Aka, just brute force it. It has a 21.5% chance of being outside the circle every time :)

Answer 2

Generate random numbers for the x and y component of the vector such that x < rectangle.width and y < rectangle.heigth. Then check whether x^2 + y^2 < circle.radius^2 and throw the vector away if so.

Update: Another way which generates a non uniform distribution but doesn't throw away any vector is as follows: Randomly choose an angle from the interval 0 to 2*pi. Now determine the length of the vector by randomly choosing a value from the interval determined by the intersections of the line with the former angle starting at the center with the circle and the rectangle. The resulting distribution will be uniform when protected onto the circle. It will also be uniform for each angle. It wont be uniform in the plane however.