Question

I’m working on a simple tutorial, and I’d like to randomly generate the positions of the red and green boxes in the accompanying images anywhere inside the dark gray area, but not in the white area. Are there any particularly elegant algorithms to do this? There are some hackish ideas I have that are really simple (continue to generate while the coordinates are not outside the inside rectangle, etc.), but I was wondering if anyone had come up with some neat solutions.

Thanks for any help!

Answer 1

Simplicity is a sort of elegance in its own right, so I agree with Jon: take a Monte Carlo approach and continue sampling until you get a valid value.

If you wanted to guarantee that you’d never place the red or green squares inside the white box, you could use the following simple algorithm:

• Determine the height h_S and width w_S of the square you’re placing.

• Divide the gray area into 8 rectangular regions R = {R₁, R₂, … R₈}, defined by the white box. (Imagine a tic-tac-toe grid with the white box at the center; this defines the surrounding eight regions.)

• Let P(S is placed in R_i) = A(R_i) / A(R), where A(R_i) is the area in which the center of S can be placed: that is, a region which is of area (h_Ri – h_S) · (w_Ri – w_S).

• Select a region according to the above probabilities. Then select a point in that region from a uniform distribution of the available x- and y-coordinates.

• Done!