Simple Curve Embedding

Given a curve f and a surface S, how hard is it to find a simple curve f ′ ⊆ S that is the most similar to f? We introduce and study this simple curve embedding problem for piecewise linear curves and surfaces in R and R, under Hausdorff distance, weak Fréchet distance, and Fréchet distance as similarity measures for curves. Surprisingly, while several variants of the problem turn out to have polynomial-time solutions, we show that in R the simple curve embedding problem is NP-hard under Fréchet distance even if S is a plane, as well as under weak Fréchet distance if S is a terrain. Additionally, these results give insight into the difficulty of computing the Fréchet distance between surfaces, and they imply that the partial Fréchet distance between non-planar surfaces is NP-hard as well.