Several numerical algorithms have been developed in the literature and employed for curves reconstruction. However, these techniques are developed within the discrete setting, namely the super-resolved image is defined on a finer grid than the observed images. Conversely, off-the-grid (or gridless) optimisation does not rely on a fine grid and offer a tractable theoretical and numerical framework. In this work, we present a gridless method accounting for the reconstruction of both open and closed curves, based on the latest theoretical development in off-the-grid curve reconstruction.