sborník

Edita Vranková
From all previous statements it follows that a system of polygons is
a dense periodical placement Mup of polygon M along a line with
symmetries if and only if is a dense two-periodical placement MM'u along a
line with basis u [11, Cor. 2], that Mup=MM'u. This equivalence between
dense placements Mup and MM'u is important for construction of the dense
two-periodical placement MM'u of the polygon M from the optimization
viewpoint. The construction of the dense two-periodical placement MM'u
corresponds to the (second) way for creation direct alternate layout for
stamping, which is mentioned at the beginning of this section.
A (total) dense two-periodical placement MM'u=Uu fulfils these
conditions (Fig. 6):
• The polygons M and M(u) are densely placed, that is u∈D(M,M).
• The polygons M and M′(v) are densely placed, that is v∈D(M′,M).
• The polygons M(u) and M′(v) are densely placed, that is
v∈(D(M′,M(u)).
Hence u∈D(M,M) and v∈ I(u)=D(M′,M)∩(D(M′,M(u)). We obtain the
basis u=u−p of total dense two-periodical placement by determining the
point u. This point is one of intersections of an half-line or with the set
D(M,M) [10], where r is an arbitrary vector collinear with the strip axis
(Fig. 6). The intersection of D(M′,M) and D(M′,M(u)) contains, in general, a
finite number of points. For rectangular axial polygons, the set I(u)
contains an union of broken line and points, which are common to both sets
(it does not need to contain a regular points, I R (u)⊂I(u)). We choose the
reference point v of polygon M′(v) from the mentioned intersection.
D(M′,M)
M
M′
a
o
d
b=v
p
u
c
D(M′,M(u))
Fig. 6: To algorithm for construction of (total) dense placement
The enter date for algorithm are the polygon M and point o. The steps of
the algorithm can be described briefly as follows:
1. Construct the set D(M,M) and select of vector r parallel to the strip axis.
u
e
D(M,M)
M′(v)
M(u)
r
strip
axis
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