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We construct a continuously differentiable curve in the plane that can becovered by a collection of lines such that every line intersects the curve at asingle point and the union of the lines has Hausdorff dimension 1. We show thatfor twice differentiable curves this is impossible. In that case, the union ofthe lines must have Hausdorff dimension 2. If we use only tangent lines thenthe differentiability of the curve already implies that the union of the linesmust have Hausdorff dimension 2, unless the curve is a line. We also constructa continuous curve, which is in fact the graph of a strictly convex function,such that the union of (one sided) tangent lines has Hausdorff dimension 1.

Hausdorff dimension of union of lines that cover a curve