In this paper the problem of sampling from uniform probability distributions is approached by means of space-filling curves, a topological concept that has found a number of important applications in recent years. Departing from the theoretical fact that they are surjective but not necessarily injective, the investigation focused upon the structure of the distributions obtained when their domains are swept in a uniform and discrete manner, and the corresponding values used to build histograms, that are approximations of their true PDFs. This work concentrates on the real interval [0,1] and the Sierpiński space-filling curve was chosen because of its favorable computational properties. In order to validate the results, the Kullback–Leibler and other divergence measures are used when comparing the obtained distributions in several levels of granularity with other already established sampling methods. In truth, the generation of uniform random numbers is a deterministic simulation of randomness using numerical operations. In this fashion, sequences resulting from this sort of process are not truly random. Despite this, and to be coherent with the literature, the expression “random number” will be used along the text to mean “pseudo-random number”.

在本文中，从均匀概率分布中采样的问题是通过空间填充曲线来解决的，空间填充曲线是近年来发现了许多重要应用的拓扑概念。与它们是满射但不一定是单射的理论事实不同，研究集中在以均匀和离散的方式扫描它们的域时获得的分布的结构，以及用于构建直方图的相应值，这些值是它们的近似值真正的 PDF。这项工作集中在实数区间 [0,1] 上，选择 Sierpiński 空间填充曲线是因为它具有良好的计算特性。为了验证结果，在将获得的多个粒度级别的分布与其他已经建立的采样方法进行比较时，使用 Kullback-Leibler 和其他散度度量。事实上，均匀随机数的生成是使用数值运算对随机性的确定性模拟。以这种方式，由这种过程产生的序列并不是真正随机的。尽管如此，为了与文献保持一致，文本中将使用“随机数”一词来表示“伪随机数”。