sborník

Edita Vranková
(intM∩intN=∅). The overlapping of two arbitrary polygons M(x) and N or
its dense placements can be expressed geometrically by the following sets
O(M,N)={x∈E 2 ; M(x) overlaps N, i.e. intM(x)∩intN ≠ ∅},
D(M,N)={x∈E 2 ; M(x), N are densely placed,
i.e. ∂M(x)∩∂N ≠ ∅ and intM(x)∩intN = ∅},
where M(x) is the translated position of polygon M=M(o) to the (reference)
point x and the point o (origin) is a fixed point in the plane E 2 [7]. Some
constructions of the set D(M,N) are described e.g. in [3], [6], [7]. We
suppose that the origin o is an interior point of the polygon M.
2 Two-periodical placement of rectangular
polygons along a line
Among the simplest tasks of the periodical placements belong placements
of mutually congruent polygons (M=N). In this case is the set D(M,M)
symmetrical by origin o [7]. If every two neighboring translated polygons
M i and M i+1 (i∈Z) are dense placed in strip, then we obtain dense periodical
placement of polygon M along a line [9].
Definition 1. A periodical placement of union U=M∪M'(x) nonoverlapping
polygons M and M'(x) along a line given by vector u, where
M′=s o (M) is the image of M by the central symmetry s o (with symmetry
center o) and M'(x) is the translated position of M' to the point x [7] we call
two-periodical placement of polygon M along a line with basis u (Fig. 1)
and we denote it by Uu or MM'u.
M′ −1
u
M′(x) x
M′ 1 M′ 2
o
M
u
M 1 M 2
M′
M −1
w
Fig. 1: Two-periodical placement Uu of polygon M with along a line
Analogically by [9], [10], two-periodical placement of a polygon M is
the system
Uu=MM'u={U i =U+iu; i∈Z}.
of non-overlapping figures U i which are copies of the figure
U=M∪M'(x) under translations given by the vectors iu for all integers i∈Z
(see Fig. 1). The polygons M, M'(x) are called the generators of the twoperiodical
placement MM'u. The basis u defines for plane strip (with width
w) a translation in strip. It is obvious that M=M(o)=M 0 and U i =M i ∪M' i (x).
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