Robustness of Boolean operations on subdivision-surface models

This work was presented in two parts at Dagstuhl seminar 08021. The two presentations described work in progress, including a ``backward bound'' for a combined backward/forward error analysis for the problem mentioned in the title. We seek rigorous proofs that representations of computed sets, produced by algorithms to compute Boolean operations, are well formed, and that the algorithms are correct. Such proofs should eventually take account of the use of finite-precision arithmetic, although the proofs presented here do not. The representations studied are based on subdivision surfaces. Such representations are being used more and more frequently in place of trimmed NURBS representations, and the robustness analysis for these new representations is simpler than for trimmed NURBS. The particular subdivision-surface representation used is based on the Loop subdivision scheme. The analysis is broken into three parts. First, it is established that the input operands are well-formed two-dimensional manifolds without boundary. This can be done with existing methods. Secondly, we introduce the so-called ``limit mesh'', and view the limit meshes corresponding to the input sets as defining an approximate problem in the sense of a backward error analysis. The presentations mentioned above described a proof of the corresponding error bound. The third part of the analysis corresponds to the ``forward bound'': this remains to be done.