Abstract: Evaluating curves on surfaces is a frequently occurring operation in a number of applications involving surface interrogations. For example, evaluating the intersection curve of two surfaces is critical to boundary (B-rep) computation, and in applications involving visibility and rendering, the ability to evaluate the silhouette curve of surfaces is important. While dealing with high degree surfaces, these curves usually consist of a number of components, including loops. We present a new algebraic loop characterization algorithm that can be applied in a number of applications. In particular, we discuss its application to the intersection curve of two surfaces and the silhouette curve of a surface. Unlike some other loop detection algorithms, our method can be applied even when the curve contain(s) singularities.