Triangulations have been one of main topics in computational geometry and other fields in recent years. In the planar case, any pair of triangulation can be transformed to each other by a sequence of so-called Delaunay flips, and enumeration of all triangulations can be done by reverse search. However, there is no known result for higher-dimensional triangulations. Recently, some types of triangulations have been found to bridge geometric issues and algebraic ones. Regular triangulations are of such a type, and form a meaningful wide subclass of triangulations of points in general dimensions. Especially, regular triangulations have a close connection with discriminants of polynomials in several variables. Restricting ourselves to the class of regular triangulations in any dimensions, we know that such triangulations correspond to vertices of some polytope, and we can enumerate all regular triangulations by reverse search.
