In mathematics curves are defined as the images of continuous real functions defined on closed intervals and these continuous functions are called parameterizations of the corresponding curves. If only simple curves of finite lengths are considered, then parameterizations can be restricted to the injective continuous functions or even to the continuous length-normalized parameterizations. In addition, a plane curve can also be considered as a connected one-dimensional compact subset of points. By corresponding effectivizations, we will introduce in this paper four versions of computable curves and show that they are all different. More interestingly, we show also that four classes of computable curves cover even different sets of points.