Abstract
<p>Geometric Disks Covering (GDC) is one of the most typical and well studied problems in computational geometry. Geometric disks are well known 2-D objects which have surface area with circular boundaries but differ from polygons whose surfaces area are bounded by straight line segments. Unlike polygons covering with disks is a rigorous task because of the circular boundaries that do not tessellate. In this paper, we investigate an area approximate polygon to disks that facilitate tiling as a guide to disks covering with least overlap difference. Our study uses geometry of tessellable regular polygons to show that hexagonal tiling is the most efficient way to tessellate the plane in terms of the total perimeter per area coverage.</p>
Highlights
Tiling differs from covering in that the former is a family of sets without overlap whereas the latter covers the entire plane with no gaps but with overlaps (Lessard, 2000, p.17)
Hexagonal tiling as a guide to disk covering is proved to have the least overlap difference of (2 − √3)R which is 13.4% over the diameter of the disk. This implies that regular hexagon has the minimum width and is the best geometric object for optimal disk covering in a plane A formulae for apothem r = R os (π) and total overlap difference d = 2nR *1 − cos (π)+ for tessellable regular polygon inscribed in disks for covering were put forward
The findings in this study suggest that disks covering using hexagonal tessellation offers an optimal covering area of 82.7% per disks area
Summary
Tiling differs from covering in that the former is a family of sets without overlap whereas the latter covers the entire plane with no gaps but with overlaps (Lessard, 2000, p.17). Squares and hexagons are known to be the only Archimedean tiling’s with lattice polygon (Ding, 2010, p.7). Any regular polygon that can tile has the property of covering. It is often useful to consider the single regular polygon whose area approximates that of a circle. This regular polygon could be a guide in our geometric disks covering problem
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