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
