平移和离散旋转下的泛凸覆盖问题

IF 0.5 4区数学 Q3 MATHEMATICS

Advances in Geometry Pub Date : 2022-11-27 DOI:10.48550/arXiv.2211.14807

Mook Kwon Jung, S. Yoon, Hee-Kap Ahn, T. Tokuyama

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引用次数: 0

摘要

摘要我们考虑周长为2的平面对象（或等效地，长度为2的闭合曲线）的最小面积通用覆盖，允许平移和离散旋转。特别地，我们证明了当π的平移和离散旋转被允许时，解是高度为1的等边三角形。我们还给出了长度为2的闭合曲线在π/2和2π/3的倍数的平移和离散旋转下的凸覆盖。我们证明了该覆盖的任何真闭子集都不是π/2倍数离散旋转的覆盖，π/2是一个高度小于1的等边三角形，并推测这种覆盖是最小面积的凸覆盖。最后，我们给出了所有整数k=3在2π/k的平移和离散旋转下所有单位段的最小面积凸覆盖。

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Universal convex covering problems under translations and discrete rotations

Abstract We consider the smallest-area universal covering of planar objects of perimeter 2 (or equivalently, closed curves of length 2) allowing translations and discrete rotations. In particular, we show that the solution is an equilateral triangle of height 1 when translations and discrete rotations of π are allowed. We also give convex coverings of closed curves of length 2 under translations and discrete rotations of multiples of π/2 and of 2π/3. We show that no proper closed subset of that covering is a covering for discrete rotations of multiples of π/2, which is an equilateral triangle of height smaller than 1, and conjecture that such a covering is the smallest-area convex covering. Finally, we give the smallest-area convex coverings of all unit segments under translations and discrete rotations of 2π/k for all integers k=3.

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来源期刊

Advances in Geometry 数学-数学

CiteScore

1.00

自引率

0.00%

发文量

审稿时长

>12 weeks

期刊介绍： Advances in Geometry is a mathematical journal for the publication of original research articles of excellent quality in the area of geometry. Geometry is a field of long standing-tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity; geometric ideas and geometric language permeate all of mathematics.