The simplest class of manifolds that pose nontrivial reconstruction problems are curves in the plane. We will describe two algorithms for curve reconstruction, Crust and NN-Crust in this chapter. First, we will develop some general results that will be applied to prove the correctness of the both algorithms.

A single curve in the plane is defined by a map ξ: [0, 1] → ℝ2 where [0, 1] is the closed interval between 0 and 1 on the real line. The function ξ is one-to-one everywhere except at the endpoints where ξ(0) = ξ(1). The curve is C1-smooth if ξ has a continuous nonzero first derivative in the interior of [0, 1] and the right derivative at 0 is same as the left derivative at 1 both being nonzero. If ξ has continuous ith derivatives, i ≥ 1, at each point as well, the curve is called Ci-smooth. When we refer to a curve Σ in the plane, we actually mean the image of one or more such maps. By definition Σ does not self-intersect though it can have multiple components each of which is a closed curve, that is, without any endpoint.

For a finite sample to be a ε-sample for some ε > 0, it is essential that the local feature size f is strictly positive everywhere.