sborník

ON TWO APPROACHES FOR CONSTRUCTION …
For the two-periodical placement MM'u of an (axial) polygon M along a
line with basis u is needed to make the set of type D(M,M), and the set of
type D(M′,M), too [12].
From simple properties of translations and central symmetries it results
that a central symmetry s p by point p transforms every periodical placement
Mu with basis u along a line to the periodical placement s p (Mu)=s p (M)u
along a line with the same basis u [11].
Definiton 2 ([5] or [11], Def 1). If the polygon M and its copy s p (M) by
central symmetry s p are not overlapping, then we call the union Mu∪s p (Mu)
of periodical placements Mu and s p (Mu) as a periodical placement of
polygon M along a line with symmetries and we denote it by Mup. Vector
u is called basis, the point p symmetry point and the line incident with the
point p and parallel to the vector u is called axis (of strip) of periodical
placement of the polygon M along a line with symmetries (Fig. 2).
M′
u
−1
v=s p (o) M′ 1
M′ 2
u M′(v)
w
M −1
M o u M 1
M 2
M′
Fig. 2: Periodical placement of polygon M along a line with symmetries
In the placement Mup vector v=s p (o)−o=v−o determines translation such
that polygon s p (M) = M'(v) is a copy of the polygon M′. The polygons which
occur in the placement Mup can by expressed in the form M i =M+iu or
M′ i =M′+iu+v for all i∈Z. The polygons M, M'(v) we call the generators of
the two-periodical placement.
In the next two sections we show two approaches for making of dense
two-periodical placement of rectangular polygon along a line. There are
geometrical models of two ways for creation direct alternate layout for
stamping, which are used in engineering for production of cutting pieces.
3 A non-total dence two-periodical placement of
rectangular polygons along a line
The first approach used in engineering practice consists in following: in
the first place it is cut-out a pair of cutting pieces symmetric by some point
lying at the strip axis using two blades ([1], [2]) and then are cut-out
repeatly another pairs of congruent pieces from the strip metal (during its
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