## Popov V.V. On algorithm of numbering of triangulations

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*Popov Vladimir Valеntinovich*Candidate of Physical and Mathematical Sciences, Associate Professor, Department of Computer Sciences and Experimental Mathematics, Volgograd State University

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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

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Prosp. Universitetsky, 100, 400062 Volgograd, Russian Federation

**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 P = {p

_{1},p

_{2},...,p

_{N}} is a finite set of points in three-dimensional space.

The triangulation of this set is such a sequence S

_{1}, S

_{2}, ..., S

_{k}of tetrahedrons with vertexs from P, which union is equal to convex hull conv(P) of P, and the intersection Si∩ Sj, i≠ j is or empty, or is the common vertex or the common edge or the common face of tetrahedrons S

_{i}and S

_{j}, and each point pi is the vertex of some S

_{j}.

At first, the algorithm A is described, which gives us the list of all triangulations on P, containing some tetrahedron S

_{1}with vertexs from a set of P without points from P other than his vertexs. Let at some k the list of tetrahedrons be already defined S

_{1}, S

_{2}, ..., S

_{k}which can be completed to some triangulations for P, but the union V = S

_{1}∪ S

_{2}∪ ... ∪ of S

_{k}is not equal to conv(P). Then there will be such two-dimensional face F a set V and such tetrahedron of S with vertexs from P which doesn’t contain points of a set P other than his vertexs, and V ∩ S = F. Among tetrahedrons S, which can be

constructed by this way, we choise such S, that the sequence (i

_{1},i

_{2},i

_{3},i

_{4}) of numbers of his vertex has a minimum value with respect to lexicographic order.

Now we put S

_{k+1}= S. After some such a steps we get a triangulation of P. To build other triangulations it is necessary to delete tetrahedrons with big numbers and add new tetrahedrons before receiving new triangulations.

Let G be a boundary of the set conv(P). We assume that G∩P={p

_{1},p

_{2},...,pl}, where l≥3. and segment [p

_{1},p

_{2}] doesn’t contains some points from P other than p

_{1}and p

_{2}. Let 2 < i

_{3}≤ l < i

_{4}and S is the tetrahedrons with vertexes p

_{1}, p

_{2},pi

_{3},pi

_{4}, which does not containes the points from P other than his vertexs. Applying algorithm A to S, we obtain some triangulations of a set P.

Changing i

_{3}, i

_{4}, we get all the triangulations. By this way we realies algorithm A.

Algorithms A and B allows us to receive, for example, the following results:

(1) Let P is the set of vertexs of a cube. Then the number of triangulations of P is equall 74.

(2) Let P′= P ∪{p}, where p is the point of the edge of cube. Then P′ has 276 triangulations.

**Key words:**triangulation, tetrahedron, simplex, number of triangulations, convex hull.

On Algorithm of Numbering of Triangulations by Popov V.V. is licensed under a Creative Commons Attribution 4.0 International License.

**Citation in English: Science Journal of Volgograd State University. Mathematics. Physics. №5 (24) 2014 pp. 40-45**