sborník

Edita Vranková
translations). This way is geometrically modeled by creation of non-total
dense two-periodical placement of a polygon along a line.
Definition 3. Two-periodical placement of the polygon M along a line
such that periodical placement MM'u=Uu of union M∪M'(x)=U along a line
is dense and the polygons M, M'(x) are densely placed, too we call non-total
or non-complete (see [12], Def. 2) dense two-periodical placement of
polygon M along a line with basis u (Fig. 3).
Fig. 3: A non-total dense two-periodical placement of polygon M
A non-total dense two-periodical placement MM'u = Uu fulfils these
conditions (Fig. 4):
• The polygons M a M′(v) are densely placed, that is v ∈ D(M′,M).
• The polygons U * a U * (u) are densely placed, that is u ∈ D(U * ,U * ),
where U * is the (rectangular) polygon, which is determined by external
boundary of the union M∪M'(v). We obtain the basis u=u−p of non-total
dense two-periodical placement by determining point u. This point is one of
intersection points of the half-line pr with the set D(U * ,U * ) [10], where r is
an arbitrary vector collinear with the strip axis (Fig. 4) and the point p
(lying on strip axis) is the center of the line segment ov. It is evident that
U * =U * (p).
D(M′,M)
D(U*,U*)
M′(v)
o
p
v
M
U*
p u u r
U*(u)
M′
Fig. 4: To algorithm for construction of non-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).
2. Take one point from vertexes of the set D(M′,M) and construct figure
M∪M′(v) and the (rectangular) polygon U ∗ .
296