Acute triangulations of trapezoids
type: Item
instance of: scholarly article
title: Acute triangulations of trapezoids
zbMATH Open document ID: 1205.52006
author: Liping Yuan
published in: Discrete Applied Mathematics
publication date: 2010-10-11T00:00:00Z
review text: A polygon is said to have an acute triangulation of size \(n\) (or can be triangulated by \(n\) acute triangles) if it can be covered by \(n\) acute triangles each two of which share one vertex or one edge at most. It is proved by \textit{C. Cassidy} and \textit{G. Lord} in [J. Recr. Math. 13, 263--268 (1980/1981)] that every square can be triangulated by \(8\), and no less than 8, acute triangles. This result is extended to arbitrary rectangles by \textit{T. Hangan}, \textit{J. Itoh}, and \textit{T. Zamfirescu} in [Bull. Math. Soc. Sci. Math. Roum., Nouv. Sér. 43 (91), No. 3--4, 279--285 (2000; Zbl 1048.51501)]. The paper under review proves that a trapezoid that is not a rectangle can be triangulated by 7 or less acute triangles and that there is a trapezoid that cannot be triangulated by less than 7 acute triangles. Thus a rectangle can be defined to be a trapezoid that does not have an acute triangulation of size 7 or less. The paper also talks about analogous known results pertaining to other figures such as quadrilaterals, pentagons, and Platonic surfaces. It mentions \textit{H. Maehara}'s result in [Proc. JCDCG 2000, Lecture Notes in Comput. Sci. 2098, 237--243 (2001; Zbl 0998.52005)] that every quadrilateral can be triangulated by 10 or less acute triangles and that there is a quadrilateral, albeit non-convex, for which 10 is minimal, and raises the question whether every \textit{convex} quadrilateral can be triangulated by 8 or less acute triangles.
reviewed by: Mowaffaq Hajja
Mathematics Subject Classification ID: 52B05
Mathematics Subject Classification ID: 52C20
zbMATH DE Number: 5798505
zbMATH Keywords: triangulation
zbMATH Keywords: cover
zbMATH Keywords: acute triangulation
zbMATH Keywords: trapezoid
MaRDI profile type: Publication
full work available at URL: j.dam.2010.02.008
OpenAlex ID: W2089218955
cites work: Q3278733
cites work: Q4814892
cites work: Acute triangulations of flat tori
cites work: Acute triangulations of the regular dodecahedral surface
cites work: Acute triangulations of the regular icosahedral surface
cites work: Q4547812
cites work: Acute triangulations of polygons
cites work: Q3007947
cites work: Q3402528
cites work: Acute triangulations of flat Möbius strips
cites work: Q5705828
DOI: 10.1016/J.DAM.2010.02.008
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description: scientific article; zbMATH DE number 5798505