We report on an adapted version of the Bentley-Ottmann algorithm for computing all the intersection points among the edges of the projection of a three-dimensional graph. This graph is given as a set of vertices together with their space Euclidean coordinates, and a set of edges connecting them. More precisely, the three-dimensional graph represents the approximation of a closed and smooth implicitly defined space algebraic curve, that allows us a simplified treatment of the events encountered in the Bentley-Ottmann algorithm. As applications, we use the adapted algorithm to compute invariants for each singularity of a plane complex algebraic curve, i.e. the Alexander polynomial, the Milnor number, the delta-invariant, etc.