2022/11/27 by Jung, Mook Kwon, Yoon, Sang Duk, Ahn, Hee-Kap +1
#05B40 #52C15 #Computational Geometry (cs.CG) #F.0 #FOS: Computer and information sciences #G.0
paper · doi:10.48550/arxiv.2211.14807
We consider the smallest-area universal covering of planar objects of perimeter 2 (or equivalently closed curves of length 2) allowing translation and discrete rotations. In particular, we show that the solution is an equilateral triangle of height 1 when translation and discrete rotation of π are allowed. Our proof is purely geometric and elementary. We also give convex coverings of closed curves of length 2 under translation and discrete rotations of multiples of π/2 and 2π/3. We show a minimality of the covering for discrete rotation of multiples of π/2, which is an equilateral triangle of height smaller than 1, and conjecture that the covering is the smallest-area convex covering. Finally, we give the smallest-area convex coverings of all unit segments under translation and discrete rotations 2π/k for all integers k≥ 3.