Given a finite set of points P ? R3, sampled from a surface S, surface reconstruction problem computes a model of S from P, typically in the form of a triangle mesh. The problem is ill-posed as various models can be reconstructed from a given point set. In this paper, curve reconstruction in R2, is initially looked at using the Delaunay triangulation (DT) of a point set. The key idea is that the edges in the DT are prioritized and the interior or exterior edges of the DT are removed as long as it has at least one adjacent triangle. Theoretically, it is shown that the reconstruction is homeomorphic to a simple closed curve. Extending this to 3D, an approach based on ‘retaining solitary triangles’ and ‘removing triangles anywhere’ has been proposed. An additional constraint based on the circumradius of a triangle has been employed. Results on public and real-world scanned data, and qualitative/quantitative comparisons with existing methods show that our approach handles diverse features, outliers and noise better or comparable with other methods. © 2017 Association for Computing Machinery.