Generalized closing theorems
type: Item
instance of: scholarly article
title: Generalized closing theorems
zbMATH Open document ID: 1235.51022
DOI: 10.4171/EM/177
published in: Elemente der Mathematik
publication date: 2011-12-29T00:00:00Z
review text: Closing theorems, in particular the theorem of Poncelet, are results of projective geometry. The author formulates and proves a generalized closing theorem for families of Euclidean spheres in \(\mathbb{R}^d\). A corollary of this theorem gives a general closing principle for spheres in \(\mathbb{R}^3\), from which the classical theorems in \(\mathbb{R}^2\) of Poncelet, Steiner and Emch, and the zigzag theorem can be derived. Further results are the extension of the above mentioned classical theorems to \(\mathbb{R}^d\) and an alternative, elementary proof of the Emch therorem.
reviewed by: Dimitris P. Vartziotis
Mathematics Subject Classification ID: 51M04
zbMATH DE Number: 5992940
zbMATH Keywords: closing theorem
zbMATH Keywords: Poncelet theorem
zbMATH Keywords: Emch theorem
zbMATH Keywords: Steiner theorem
zbMATH Keywords: zigzag theorem
author: Vladimir Yu. Protasov
MaRDI profile type: Publication
cites work: Poncelet theorems
cites work: Modular curves and Poncelet polygons
cites work: A Theorem About Zig-Zags Between Two Circles
cites work: On Cayley's explicit solution to Poncelet's porism
cites work: Poncelet-type problems, an elementary approach
cites work: Q3372927
cites work: Q3706091
full work available at URL: https://doi.org/10.4171/em/177
OpenAlex ID: W2169667933
description: scientific article; zbMATH DE number 5992940