Abstract: It is a necessary condition that the curve is continuous and sufficient smooth when we solve the differential equation defined on the curve in engineering calculation. This paper discusses the smooth curve approximation for non-smooth curves with the same length on local interval. We prove that the following theoretical results based on the intermediate value theorem: if the measurable length of non-smooth points in a continuous curve has limit length, on any local sub-intervals, there always existences at least one smooth spline curve can be used to approximate the curve with same length as long as the length can be measured on this fixed sub-intervals. Furthermore, if a reference point is given, the unique smooth spline curve can be derived easily. Based on these theoretic results, we also propose the algorithm and present some numerical results to show the simplicity and efficiency of this method, which leads to the optimal solution for the non-smooth curve approximation.

Key words: measurable length, intermediate value theorem, smooth curve, same length app-roximation