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 = {p1,p2,...,pN} is a finite set of points in three-dimensional space.
The triangulation of this set is such a sequence S1, S2, ..., Sk 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 Si and Sj, and each point pi is the vertex of some Sj.
At first, the algorithm A is described, which gives us the list of all triangulations on P, containing some tetrahedron S1 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 S1, S2, ..., Sk which can be completed to some triangulations for P, but the union V = S1∪ S2∪ ... ∪ of Sk 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 (i1,i2,i3,i4) of numbers of his vertex has a minimum value with respect to lexicographic order.
Now we put Sk+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={p1,p2,...,pl}, where l≥3. and segment [p1,p2] doesn’t contains some points from P other than p1and p2. Let 2 < i3≤ l < i4 and S is the tetrahedrons with vertexes p1, p2,pi3,pi4, 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 i3, i4, 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.
Let P = {p1,p2,...,pN} is a finite set of points in three-dimensional space.
The triangulation of this set is such a sequence S1, S2, ..., Sk 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 Si and Sj, and each point pi is the vertex of some Sj.
At first, the algorithm A is described, which gives us the list of all triangulations on P, containing some tetrahedron S1 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 S1, S2, ..., Sk which can be completed to some triangulations for P, but the union V = S1∪ S2∪ ... ∪ of Sk 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 (i1,i2,i3,i4) of numbers of his vertex has a minimum value with respect to lexicographic order.
Now we put Sk+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={p1,p2,...,pl}, where l≥3. and segment [p1,p2] doesn’t contains some points from P other than p1and p2. Let 2 < i3≤ l < i4 and S is the tetrahedrons with vertexes p1, p2,pi3,pi4, 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 i3, i4, 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
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