We are currently studying the behavior of algorithms in the neighborhood of singular curves; we are developing algorithms that detect and locally resolve the geometric structure of singular curves in a neighborhood of the singularities. We do not know of any algorithms that can carry out this task for level sets of generic functions.
We hope to extend this study to provide users with the first general-purpose algorithm that correctly identifies and depicts singular surfaces. We are using ideas from Mike Henderson at IBM's T. J. Watson Research Center to construct an algorithm that computes implicitly-defined (nonsingular) surfaces.
In the future we will continue our efforts to provide users with high-level mechanisms to compute and graphically display level curves and surfaces by means of a Pisces command language. We will concentrate on communication between algorithms, so that Pisces becomes an extensible platform for computing and visualizing implicitly-defined curves and surfaces.