# Discrete approximation to a continuous probability density function

I want to approximate a continuous, finite probability density function, with a specified number $N$ of points, in the following way:

If the pdf is 1-dimensional, defined over the section [0,1], then I want to put $N$ points in that section, so that the cdf between each two adjacent points is the same. This is easy - just put the first point at 0, and put each subsequent point such that the integral of the pdf between the previous point to the current point is $1/(N-1)$.

My actual question is about a 2-dimensional pdf, defined over the square [0,1]x[0,1]. I want to put $N$ points in that square, so that the cdf in each square that touches two points (and does not contain any point in the interior) is the same. How can I arrange the points?

(As a usage example, suppose I have a square grayscale picture, such that each point has a brightness value. However, I need to print this picture using only black points. So, I want to arrange the black points such that their density approximates the level of brightness).

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For the usage example, there are several algorithms that will surely beat anything that someone can come up with on the spot. –  ˈjuː.zɚ79365 Jun 9 '13 at 5:40