sborník

ON TWO APPROACHES FOR CONSTRUCTION …
2. Determine and take of point u∈or ∩D(M,M) and put vector u=u−o.
3. Construct the sets D(M′,M) and D(M′,M(u)).
4. Compute the intersection I(u)=D(M′,M)∩D(M′,M(u)).
5. Take one point from the set I(u) and put vector v=v−o.
6. The two polygons M, M'(v) generate the two-periodical placement Uu.
R e m a r k. The set D(M′,M(u)) is congruent to the set D(M′,M) and it is
a copy of the set D(M′,M) under translation given by the vector u. Dense
two-periodical placement MM'u=Uu can be made as the set all copies of the
union U=M∪M'(v) under all translations iu for i∈Z.
Complexity of this algorithm depends again on the complexity of the
constructions D(M,M) and D(M′,M) and on the complexity of the
determining of the intersection or∩D(M,M) and I(u). It can be expected
that we obtain the optimal (total) dense two-periodical placement either for
some regular point from the set I R (u) or for some point from the set I(u).
5 Conclusions
For the current stage of production in many branches of industry it is typical
the robotization and automation not only of the whole production process,
but also in process technological preparation of procedures, too. One of
technological preparation is creation of (optimal) cutting planes. These
problems are very current at present times above all from minimizing of
production costs viewpoint, especially in connection with important
development of the automobile industry in Slovakia.
The two presented constructions for dense two-periodical placement of
rectangular polygon M along a line resulted from theoretical research with
respect to applications in engineering in the sheet metal cutting process (the
production of rectangular cutting pieces) - in solving (optimal) direct
alternate layout for stamping. We will study and publish the criterions and
conditions of optimization for both described ways in future.
The reviewed cutting plans are used often in engineering practice,
especially in production of rectangular pieces with smaller dimensions and
for bigger series. The tools with bigger number blades have shortest utility,
the rotation of strip impedes for a change operate of tool [2]. A choice on
procedure for mentioned direct alternate layout depends consequently on
concrete demands of production, material and technological conditions.
Acknowledgement
The research was supported by the grant VEGA No 1/2006/05.
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