Robustness of Boolean operations on subdivision-surface models

Abstract

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.