Optimising triangulated polyhedral surfaces with self-intersections

We discuss an optimisation procedure for triangulated polyhedral surfaces (referred to as (2 - 3) D triangulations) which allows us to process self-intersecting surfaces. As an optimality criterion we use minimisation of total absolute extrinsic curvature (MTAEC) and as a local transformation - a diagonal flip, defined in a proper way for (2 - 3) D triangulations. This diagonal flip is a natural generalisation of the diagonal flip operation in 2D, known as Lawson's procedure. The difference is that the diagonal flip operation in (2 - 3)D triangulations may produce self-intersections. We analyze the optimisation procedure for (2 - 3) D closed triangulations, taking into account possible self-intersections. This analysis provides a general insight on the structure of triangulations, allows to characterise the types of self-intersections, as well as the conditions for global convergence of the algorithm. It provides also a new view on the concept of optimisation on the whole and is useful in the analysis of global and local convergence for other optimisation algorithms. At the end we present an efficient implementation of the optimality procedure for (2 - 3)D triangulations of the data, situated in the convex position, and conjecture possible results of this procedure for non-convex data.