Random point in a circle
You’re given a procedure that with a uniform probability distribution outputs random numbers between 0 and 1 (to some sufficiently high degree of precision, with which we need not concern ourselves in this puzzle). Using a bounded number of calls to this procedure, construct a procedure that with a uniform probability distribution outputs a random point within the unit circle.
The problem does not state whether the random number is inclusive, exclusive or some combination, of the bounds 0 and 1. My answer assumes the typical random number generator, which is inclusive of 0 and exclusive of 1. It takes exactly two calls to the random number generator.
radius = random / 2
degrees = random * 360
x = radius * sin(degrees)
y = radius * cos(degrees)
Now the problem with this solution is that it lacks uniform distribution. Half the points will have radius < 0.25 and half 0.25 to 0.5, but the area in the circle with radius < 0.25 is only 25% of the total area of the circle. So we need to distribute the points along the radius using the inverse square of the random value so that the probability of a point having radius r is proportional to the area of a circle with radius r (i.e. a radius that gives a circle with area a is half as likely as a radius that gives a circle with area 2a).
radius = sqrt(random) / 2
degrees = random * 360
x = radius * sin(degrees)
y = radius * cos(degrees)
Your Answer
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