Densest translational lattice packing of non-convex polygons (extended abstract)

Abstract

A translation lattice packin9 of k polygons/'1, P2,/'3,..., Pk is a (non-overlapping) packing of the k polygons which can be replicated without overlap at each point of a lattice iovo + i~v~, where vo and vt are vectors generating the lattice and io and il range over all integers. A densest translational lattice packing is one which minimizes the area [vo x vl I of the fundamental parallelogram. An algorithm and implementation is given for densest translation lattice packing. This algorithm has useful applications in industry, particularly clothing manufacture.

Stochastic refinement of the visual hull to satisfy photometric and silhouette consistency constraints

“…We generally have over 200 unfolded groups. The translational polygon packing problem is known to be NP-hard [15]; therefore, heuristic solutions are employed that trade optimality for speed.…”

“…This step is repeated a number of times, and the translation producing the smallest bounding box is used to add the group to the packing. An alternative algorithm was proposed in [15], and its evaluation in our application remains a topic for future investigation.…”