The Peano and the Hilbert curves, denoted by P and H respectively, are historically the first and some of the best known space-filling curves. They have a fractal structure, many variants (as the well-known Moore curve M or a probably new “looped” version \({\overline{H}}\) of H), and a huge number of applications in the most diverse fields of mathematics and experimental sciences. In this paper, we employ a recently proposed computational system, allowing numerical calculations with infinite and infinitesimal numbers, to investigate the behavior of such curves and to highlight the differences with the classical treatment. In particular, we perform several types of computations and give many examples based not only on the curves H and P, but also on their d-dimensional versions \(H^d\) and \(P^d\), respectively. Following our approach, it is easy to apply this new computational methodology to many other geometrical contexts, with interesting advantages such as summarizing in a single (infinite) number, representing the final result of a sequence of computations, much information both on the geometrical meaning of such a sequence and on the base geometrical structure itself.

中文翻译：

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无限数值计算应用于希尔伯特，皮亚诺和摩尔曲线

历史上分别由P和H表示的Peano和Hilbert曲线是第一条和一些最著名的空间填充曲线。他们有一个分形结构，许多变种（如著名的摩尔定律曲线中号或可能新的“循环”版\（{\划线{H}} \）的^ h），以及在数学和实验科学的最广泛领域中的大量应用。在本文中，我们采用了最近提出的计算系统，该计算系统允许使用无穷和无穷小数进行数值计算，以研究此类曲线的行为并突出经典处理的差异。特别是，我们执行几种类型的计算并不仅基于曲线H和P，而且还基于其d维版本\（H ^ d \）和\（P ^ d \）给出许多示例。， 分别。按照我们的方法，很容易将这种新的计算方法应用于许多其他几何环境，并具有有趣的优势，例如汇总一个（无限）数字，代表一系列计算的最终结果，以及有关几何意义的大量信息这样的顺序，并基于其本身的几何结构。