Efficient representation in spaces of plane curves

Abstract

This paper evaluates the Blum medial axis representation of embeddings of $S^1$ into $R^2$ from the perspective of efficiency, using a $C^1$-type metric. For compact classes of curves with Lipschitz tangent angle, we compute the $\epsilon$-entropy and compare that efficiency benchmark with uniform approximation using the Blum medial axis. In the compact setting, the boundary curve is more efficient. For noncompact classes of embeddings, we establish a geometric criterion for when the medial axis will be more efficient in an adaptive approximation.