On algorithm of numbering of triangulations

Abstract. In [1] the algorithm of numbering of all triangulations of a finite set on the plane is offered. This paper describes the modification of this algoritm for subset of points in three-dimensional space. Let ?? = {??1, ??2,..., ????} is a finite set of points in three-dimensional space. The triangulation of this set is such a sequence ??1, ??2,..., ???? of tetrahedrons with vertexs from ??, which union is equal to convex hull conv(??) of ??, and the intersection ???? ? ????, ?? ?= ?? is or empty, or is the common vertex or the common edge or the common face of tetrahedrons ???? and ????, and each point ???? is the vertex of some ????. At first, the algorithm ?? is described, which gives us the list of all triangulations on ??, containing some tetrahedron ??1 with vertexs from a set of ?? without points from ?? other than his vertexs. Let at some ?? the list of tetrahedrons be already defined ??1, ??2,..., ???? which can be completed to some triangulations for ??, but the union ?? = ??1 ? ??2 ?... ? ???????? is not equal to conv(P). Then there will be such two-dimensional face ?? a set ?? and such tetrahedron of ?? with vertexs from ?? which doesn't contain points of a set ?? other than his vertexs, and ?? ? ?? = ??. Among tetrahedrons ??, which can be constructed by this way, we choise such ??, that the sequence (??1, ??2, ??3, ??4) of numbers of his vertex has a minimum value with respect to lexicographic order. Now we put ????+1 = ??. After some such a steps we get a triangulation of ??. To build other triangulations it is necessary to delete tetrahedrons with big numbers and add new tetrahedrons before receiving new triangulations. Let ?? be a boundary of the set conv(??). We assume that ?? ? ?? = = {??1, ??2,..., ????}, where ?? ? 3. and segment [??1, ??2] doesn't contains some points from ?? other than ??1 and ??2. Let 2