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Consider the curve C = {(t, f(t): 0 < t < 1}, where f is absolutely continuous on [0, 1]. Then C has finite length, and if Ue is the &neighborhood of f in the uniform norm, we compare the length of the shortest path in UE with the length of f. Our main result establishes necessary and sufficient conditions on f such that the difference of these quantities is of order e as e -* 0. We also include a result for surfaces.

Some remarks on the variation of curve length and surface area