sborník

ON TWO APPROACHES FOR CONSTRUCTION …
3. Create the set D(U ∗ ,U ∗ ) and select of vector r parallel to the strip axis.
4. Determine the intersection and take of point u∈pr∩D(U ∗ ,U ∗ ) and put
u=u−p.
With respect to geometrical structure [7] of the set D(M,N) it can hold
for the point v according to [8] the following:
a) Point v is a vertex of a non-convex interior angle (with size 270°) of
the rectangular polygon bounded by the main part of the set D(M′,M).
b) The point v is a vertex of a convex interior angle of some rectangular
polygon bounded by a simple closed broken line contained in the set
D(M′,M) and different from the main part of D(M′,M).
c) The point v is an end-point of a simple non-closed broken lines
(including line segments) contained in the set D(M′,M).
d) The point v is an isolated point of the set D(M′,M).
Complexity of this algorithm depends on the complexity of the
constructions of the sets D(M′,M) and D(U ∗ ,U ∗ ) and on the complexity of
the determining of the intersection pr∩D(U ∗ ,U ∗ ). We expect that an optimal
non-total dense two-periodical placement we obtain for a point from one of
the eventualities a) - d).
4 A total dence two-periodical placement of
rectangular polygons along a line
The second approach which is most often used in engineering practice
consists in the following: in the first place are cut-out repeatly congruent
cutting pieces in the one line of the metal strip (during its translation) using
only one blade and then are cut-out the pieces in the second line using the
same blade after the rotation of the metal strip by 180°.
Definition 4. Two-periodical placement MM'u=Uu of the polygon M
along a line such that periodical placement Mu of the polygon M along
a line with the basis u is dense and periodical placement M'(x)u of the
polygon M'(x) along a line with the basis u is dense and the polygons M,
M'(x) are densely placed, too we call total (or complete) dense twoperiodical
placement of polygon M along a line with basis u (Fig. 5).
Fig. 5: A total dense two-periodical placement of polygon M along a line
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