In this work we propose a new method for silhouette extraction of a point cloud, via local reconstruction of a surface described implicitly by a polynomial function. This reconstruction is based on the Gradient one fitting and Ridge regression methods. The curve silhouette is implicitly defined by a system of nonlinear equations, and is obtained using numerical continuation. As a result, we observe that our method is suitable to handle noisy data. In addition, we present a method for extracting Laplacian Lines of a point cloud based on local reconstruction using the Delaunay triangulation.