  Ten years ago, Beresnevich-Dickinson-Velani initiated a project that develops
the general Hausdorff measure theory of dual approximation on non-degenerate
manifolds. In particular, they established the divergence part of the theory
based on their general ubiquity framework. However, the convergence counterpart
of the project remains wide open and represents a major challenging question in
the subject. Until recently, it was not even known for any single
non-degenerate manifold. In this paper, we settle this problem for all curves
in $\mathbb{R}^2$, which represents the first complete theory of its kind for a
general class of manifolds.
