sborník

25. KONFERENCE O GEOMETRII A POČÍTAČOVÉ GRAFICE
Edita Vranková
ON TWO APPROACHES FOR
CONSTRUCTION OF DIRECT ALTERNATE
LAYOUT FOR STAMPING
Abstract
In the presented paper we study two approaches for making of direct
alternate layout for stamping from geometrical point of view, which are
used above all in engineering in sheet metal forming for production of
cutting pieces. There are solving non-total dense and (total) dense twoperiodical
placements of rectangular and mutually congruent axial
polygons along a line.
Keywords
Cutting plan, direct alternate layout for stamping, congruent rectangular
and axial polygons, non-total and total dense two-periodical placement.
1 Introduction
Not only in engineering, but also in clothing and shoe industry are solving
tasks which lead in principle to interactive or automatic placements of plane
geometric figures without overlapping into some plane domain when only
translations of figures are allowed. An important part of technological
preparation of procedures is the construction of (optimal) cutting plan [1],
[2]. Above all in engineering practice (automobile industry - sheet metal
forming) is often solved direct alternate layout for stamping [2].
Geometrically speaking, it is a periodical placement of union of two
congruent rectangular axial polygons [9] symmetric by some point lying at
the axis of the strip along a line. We called it two-periodical placement of
rectangular axial polygons along a line.
An useful theoretical tool for solving such tasks can be the set D(M,N)
of dense placements for moving polygon M and fixed polygon N. During
placement of polygons it is necessary to provide the non-overlapping of any
two different polygons. Two polygons M, N are overlapping if they have
a common interior point (intM∩intN≠∅). Among all placements of
polygons are important their dense placements from optimization
viewpoint. The polygons M, N are dense placed if there is a common
boundary point (∂M∩∂N≠∅) and if the polygons M, N are not overlapping
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