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Let $X_0, \widetilde{X}$ be two smooth, closed and locally convex curves inthe plane with same winding number. A curvature flow with a nonlocal term isconstructed to evolve $X_0$ into $\widetilde{X}$. It is proved that this flowexits globally, preserves both the local convexity and the elastic energy ofthe evolving curve. If the two curves have same elastic energy then thecurvature flow deforms the evolving curve into the target curve $\widetilde{X}$as time tends to infinity.

Whitney-Graustein Homotopy of Locally Convex Curves via a Curvature Flow