Abstract We study the computational complexity of the Hausdorff distance of two curves on the two-dimensional plane, in the context of the Turing machine-based complexity theory of real functions. It is proved that the Hausdorff distance of any two polynomial-time computable curves is a left- Σ 2 P real number. Conversely, for any tally set A in Σ 2 P , there exist two polynomial-time computable curves such that set A is computable in polynomial time relative to the Hausdorff distance of these two curves. More generally, we show that, for any set A in Σ 2 P , there exist two polynomial-time computable curves such that set A is computable in polynomial time relative to the Hausdorff distances of subcurves of these two curves.