On Flips in Polyhedral Surfaces: a new development

Let V be a finite point set in 3D and let ST(V ) be the set of closed triangulated polyhedral surfaces with a vertex set V. Those surfaces can be dened as 2:5D (closed) triangulations of the given discrete data set V. We generalise the operation of diagonal flip for 2:5D triangulations by omitting the usual restriction that the flip operation should not produce a self-intersecting triangulation. We denote this flip operation by EDF (extended diagonal flip). Among all possible 2:5D triangulations with the vertex set V we first single out those that are topologically equivalent to the 2D sphere. We show that any two such 2:5D triangulations (if V is situated in general position), are equivalent under EDF, i.e., they can be transformed into each other via a finite sequence of EDF