Untangling Triangulations Through Local Explorations

Pankaj K. Agarwal, Bardia Sadri, and Hai Yu

In many applications it is often desirable to maintain a valid mesh within a certain domain that deforms over time. During a period for which the underlying mesh topology remains unchanged, the deformation moves the vertices of the mesh and thus potentially turns a mesh invalid, or as we call it, tangled. We introduce the notion of locally removable region, which is a certain tangled area in the mesh that allows for local removal and re-meshing. We present an algorithm that is able to quickly compute, through local explorations, a minimal locally removable region containing a seed tangled region in the invalid mesh. By re-meshing within this area, the seed tangled region can then be removed from the mesh without introducing any new tangled region. The algorithm is output-sensitive in the sense that it never explores outside the output region. Our algorithm exploits several novel insights into the structure of the tangled mesh, which may be of independent interest in contexts beyond mesh untangling.

Symposium on Computational Geometry (SoCG), 2008 [PDF]