平面常宽凸体的构造

数学物理学报 ›› 2021, Vol. 41 ›› Issue (1): 15-28.

平面常宽凸体的构造

张德燕*(),段博韬

淮北师范大学数学科学学院安徽淮北 235000

收稿日期:2019-12-19 出版日期:2021-02-26 发布日期:2021-01-29
通讯作者: 张德燕 E-mail:zhangdy8005@126.com
基金资助:
安徽省自然科学基金(1908085MA05);安徽省高校自然科学研究项目(KJ2019A0590);高校优秀人才支持计划重点项目(gxyqZD2020022)

Construction of the Planar Bodies with Constant Width

Deyan Zhang*(),Botao Duan

School of Mathematical Sciences, Huaibei Normal University, Anhui Huaibei 235000

Received:2019-12-19 Online:2021-02-26 Published:2021-01-29
Contact: Deyan Zhang E-mail:zhangdy8005@126.com
Supported by:
Supported by the NSF of Anhui Province(1908085MA05);the Natural Science Research Project of Universities in Anhui Province(KJ2019A0590);the Excellent Talents Fund Program of Higher Education Institutions of Anhui Province(gxyqZD2020022)

摘要/Abstract

摘要：

该文首先定义了一类平面曲线"杠杆轮"与它的臂函数，并利用臂函数给出杠杆轮的参数表示.其次，证明了杠杆轮是平面常宽曲线的一种等价刻画.最后，表明Reuleaux多边形是臂函数为分段常函数的一类杠杆轮，进而构造出偶数边的Reuleaux多边形.

关键词: 杠杆轮, 臂函数, 常宽曲线, Reuleaux多边形

Abstract:

Firstly, a class of planar curves "lever wheel" and their arm functions are defined, and the parameter representation of the lever wheel is established in this paper. Secondly, it is shown that the lever wheel is an equivalent characterization of the constant width curve. Finally, it is proven that the Reuleaux polygons are a class of lever wheels with piecewise constant arm functions, and Reuleaux polygons with even edges are constructed.

Key words: Lever wheel, Arm function, Constant width curve, Reuleaux polygon

中图分类号:

O174.13

张德燕,段博韬. 平面常宽凸体的构造[J]. 数学物理学报, 2021, 41(1): 15-28.

Deyan Zhang,Botao Duan. Construction of the Planar Bodies with Constant Width[J]. Acta mathematica scientia,Series A, 2021, 41(1): 15-28.

图/表 4

参考文献 9

1	徐文学.新的常宽凸集.重庆:西南大学, 2013
	Xu W X. On New Convex Set of Constant Width. Chongqing:Southwest University, 2013
2	Meissner E . Über Punktmengen konstanter Breite. Vierteljahresschr Naturforsch Ges Zürich, 1911, 56: 42- 50
3	Blatter C . Über Kurven konstanter Breite. Elem Math, 1981, 36 (5): 105- 115
4	Fujiwara M . Analytic proof of Blaschke's theorem on the curve of constant breadth with minimum area. Proceedings of the Japan Academy, 1927, 3 (6): 307- 309 doi: 10.3792/pia/1195581847
5	Evans M H . A direct proof of a theorem of Blaschke and Lebesgue. The Journal of Geometric Analysis, 2002, 12 (1): 81- 88 doi: 10.1007/BF02930861
6	潘生亮. 切线极坐标的一个应用. 华东师范大学学报(自然科学版), 2003, 1: 13- 16
	Pan S L . An application of the polar tangential coordinate. Journal of East China Normal University (Natrual Science Edition), 2003, 1: 13- 16
7	Robert T L , Qudet E . Bodies of constant width in arbitrary dimension. Math Nachr, 2007, 280 (7): 740- 750 doi: 10.1002/mana.200510512
8	徐文学, 周家足, 陈方维. 一类常宽"等腰梯形". 中国科学:数学, 2011, 41 (10): 855- 860 doi: 10.3969/j.issn.0253-2778.2011.10.002
	Xu W X , Zhou J Z , Chen F W . On the isosceles trapezoids of constant width. Chinese Science:Mathematics, 2011, 41 (10): 855- 860 doi: 10.3969/j.issn.0253-2778.2011.10.002
9	Schneider R . Convex Bodies:the Brunn-Minkowski Theory (2nd ed) New York: Cambridge University Press, 2014