The objective of this work is to investigate a new method to discover and describe all features of an unknown algebraic curve resulting in the context of surface to surface intersections. Our method involves symbolic derivation of the equation of the curve, using algebraic geometry techniques, and its representation in the Bernstein basis within a rectangular window, which allows piecewise representation and provides geometrical significance to the underlying coefficients. Novel methods to compute the characteristic points of the curve, useful in guaranteeing correct topology, are proposed. These are based on symbolic computation, adaptive methods for large scale series summation, to enhance accuracy, and minimization techniques. Interrogation of the curve, based on non-uniform adaptive subdivision, and derivation of point and parametric approximations of the implicit curve, with prescribed distance errors, based on the properties of the Bernstein representation, are also proposed for study. The significance of the proposed work is that it promises to provide a robust surface to surface intersection capability, which is necessary in a more automated mode of design, analysis and manufacturing of complex artifacts. Many elements of our technique are also parallelizable which is an important attribute for real time large scale applications.
