Densest translational lattice packing of non-convex polygons

A translational lattice packing of k polygons P"1,P"2,P"3,...,P"k is a (non-overlapping) packing of the k polygons which is replicated without overlap at each point of a lattice i"0v"0+i"1v"1, where v"0 and v"1 are vectors generating the lattice and i"0 and i"1 range over all integers. A densest translational lattice packing is one which minimizes the area |v"0xv"1| of the fundamental parallelogram. An algorithm and implementation is given for densest translational lattice packing. This algorithm has useful applications in industry, particularly clothing manufacture.

[1]  David M. Mount,et al.  The Densest Double-Lattice Packing of a Convex Polygon , 1991, Discrete and Computational Geometry.

[2]  Greg Kuperberg,et al.  Double-lattice packings of convex bodies in the plane , 1990, Discret. Comput. Geom..

[3]  Victor Milenkovic,et al.  Containment algorithms for nonconvex polygons with applications to layout , 1995 .

[4]  K. Dowsland,et al.  Solution approaches to irregular nesting problems , 1995 .

[5]  V. Milenkovic,et al.  Compaction and separation algorithms for non-convex polygons and their applications☆ , 1995 .

[6]  Victor J. Milenkovic,et al.  Multiple Translational Containment Part II: Exact Algorithms , 1997, Algorithmica.

[7]  Victor J. Milenkovic,et al.  Multiple Translational Containment Part I: An Approximate Algorithm , 1997, Algorithmica.

[8]  V. Milenkovic,et al.  Translational polygon containment and minimal enclosure using mathematical programming , 1999 .

[9]  Victor J. Milenkovic,et al.  Translational polygon containment and minimal enclosure using linear programming based restriction , 1996, STOC '96.

[10]  Zhenyu Li,et al.  Compaction algorithms for non-convex polygons and their applications , 1995 .

[11]  Paul F. Whelan,et al.  Automated packing systems: review of industrial implementations , 1993, Other Conferences.

[12]  Paul E. Sweeney,et al.  Cutting and Packing Problems: A Categorized, Application-Orientated Research Bibliography , 1992 .

[13]  Harald Dyckhoff,et al.  A typology of cutting and packing problems , 1990 .