Point-Separable Classes of Simple Computable Planar Curves

TL;DR

This paper explores four distinct classes of computable planar curves based on classical curve definitions, revealing their differences, point-separability, and the robustness of the computable length condition.

Contribution

It introduces and compares four versions of computable curves, demonstrating their differences and the point-separability of their point sets, and shows the equivalence of these classes when restricted to computable lengths.

Findings

The four classes of computable curves are all different.

These classes are point-separable in terms of the points they cover.

When restricted to computable lengths, all four classes become equivalent.

Abstract

In mathematics curves are typically defined as the images of continuous real functions (parametrizations) defined on a closed interval. They can also be defined as connected one-dimensional compact subsets of points. For simple curves of finite lengths, parametrizations can be further required to be injective or even length-normalized. All of these four approaches to curves are classically equivalent. In this paper we investigate four different versions of computable curves based on these four approaches. It turns out that they are all different, and hence, we get four different classes of computable curves. More interestingly, these four classes are even point-separable in the sense that the sets of points covered by computable curves of different versions are also different. However, if we consider only computable curves of computable lengths, then all four versions of computable curves become equivalent. This shows that the definition of computable curves is robust, at least for those of computable lengths. In addition, we show that the class of computable curves of computable lengths is point-separable from the other four classes of computable curves.