Generating a Random Point within a CircleDate: 04/02/2007 at 17:41:13 From: PC Subject: Random point within a circle. Given a circle C with center (xc, yc) and radius R, find a random point within the circle. What's the best possible way to do that in terms of computation? Date: 04/03/2007 at 08:57:09 From: Doctor George Subject: Re: Random point within a circle. Hi PC, Thanks for writing to Doctor Math. I assume that you are looking for the points to be uniformly distributed in the circle. One method is to pick random points in a square in which the circle is circumscribed. Then reject those points that are not also in the circle. Another method is to select a random pair of values (M, theta) where M is the magnitude of the vector from the center to a random point and theta is the angle by which the vector is rotated. The random points then have the following form. x = xc + M cos(theta) y = yc + M sin(theta) The distribution of theta is uniform on the interval [0, 2pi]. The distribution of M has a ramp shape on the interval [0, R]. It is a good exercise to prove that M has this distribution. To compute random values of M you can use the inverse of its cumulative distribution. Another technique is to generate a pair of values (r1, r2) that are uniform on the interval [0, R] and let M = max(r1, r2). This proof is also a good exercise. I prefer the second method, and I typically generate values of M using the (r1, r2) pairs. Does that make sense? Write again if you need more help. - Doctor George, The Math Forum http://mathforum.org/dr.math/ |
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