Question

Generate a random point within a rectangle (uniformly)

This suppose to be a simple problem.

However, in RANDOM_DATA homepage I found the following note:

However, we will not achieve uniform distribution in the simple case
of a rectangle of nonequal sides [0,A] x [0,B], if we naively scale
the random values (u1,u2) to (A*u1,B*u2). In that case, the expected
point density of a wide, short region will differ from that of a
narrow tall region. The absence of uniformity is most obvious if the
points are plotted.

I found it quite of strange… I can’t figure out why such scaling will affect the uniformity.

What am I missing?

Edit:

Thank you Patrick87 and missingno. I was searching for a theoretical reason for the statement. I now understand that the reason is not theoretical, but practical – the granularity of floating-point values.

If I’ll generate two uniform floating-points between 0 and 1 (which is an issue by itself due to the nature of floating-point value representation. Look here for an algorithm) – the granularity will be limited.

Suppose that there are X different values between 0 and 1. By scaling (u1,u2) to (u1,2*u2) we’ll have X different values in the range [0,u1] and X different values in the range [0,2*u2]. For area uniformity we should have twice as many different values in [0,2*u2] than in [0,u1].

Given that, Allow me to change my question:

How should I generate a random point within a rectangle (with uniform distribution by area)?

Answer 1

How should I generate a random point within a rectangle (with uniform distribution by area)?

This should work:

  // given A, B, dimensions of rectangle
  // given g, granularity of shorter side

  if A > B then
     bm := g
     am := floor(g * A / B)
  else then
     am := g
     bm := floor(g * B / A)

  for i := 1 to n do
     av := A * RandomInt(0..am) / am
     bv := B * RandomInt(0..bm) / bm
     print (av, bb)

EDIT: A simpler alternative would be to simply scale random floating point values by the same factor, choose points at random, and throw away points that fall outside your rectangle. However, you don’t know how many trials you’d need before you got N points in the rectangle…