Contouring 1- and 2-manifolds in arbitrary dimensions

We propose an algorithm for contouring k-manifolds (k = 1,2) embedded in an arbitrary n-dimensional space. We assume (n -k) geometric constraints are represented as polynomial equations in n variables. The common zero-set of these (n-k) equations is computed as an 1-or 2-manifold, respectively, for k = 1 or k = 2. In the case of 1-manifolds, this framework is a generalization of techniques for contouring regular intersection curves between two implicitly-defined surfaces of the form F(x,y,z) = G(x,y,z) = 0. Moreover, in the case of 2-manifolds, the algorithm is similar to techniques for contouring iso-surfaces of the form F(x, y, z) = 0, where n = 3 and only one (=3 -2) constraint is provided. By extending the Dual Contouring technique to higher dimensions, we approximate the simultaneous zero-set as a piecewise linear 1or 2-manifold. There are numerous applications for this technique in data visualization and modeling, including the processing of various geometric constraints for freeform objects, and the computation of convex hulls, bisectors, blendings and sweeps.