Poopćenje klasičnih teorema zatvaranja Ponceletovog tipa

master's thesis

• Author: Petra Zubak
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Vedran Krčadinac (član povjerenstva)
• Committee member: Ivica Nakić (član povjerenstva)
• Committee member: Eduard Marušić-Paloka (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2017-04-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Ponceletov teorem koji govori o svojstvu "zatvaranja" niza poligona koji su upisani jednoj, a opisani drugoj zadanoj konici svakako je jedan od najpoznatijih i najviše proučavanih teorema u geometriji. Poznato je još nekoliko teorema sličnog tipa, kao što su Steinerov teorem, "Zigzag" teorem i Emchov teorem koji se u stanovitom smislu može smatrati najopćenitijim među njima. U ovom radu najprije je za familiju kružnica iskazano opće svojstvo zatvaranja na nekoj kružnici \(k\) u istoj ravnini, a zatim se formulira odgovarajuće svojstvo familije sfera u \(d\)-dimenzionalnom euklidskom prostoru, u odnosu na neku kružnicu \(k\). Dokazano je da svojstvo zatvaranja na nekoj kružnici ima svaka familija \(\mathcal{F}\) takvih sfera koje uz stanovite uvjete nesingularnosti ispunjavaju jednostavnu analitičku relaciju,a središta im pripadaju jednoj kvadrici \(\Gamma\). Taj se uvjet geometrijski intepretira tako da sfere iz \(\mathcal{F}\) diraju dvije zadane sfere \(S_0\) i \(S_1\), a ortogonalne su na zadanu sferu \(Q\). Pokazuje se kako četiri klasična teorema zatvaranja slijede iz tog općeg principa zatvaranja za sfere. U završnom dijelu rada izložen je elementarni dokaz Emchovog teorema u ravnini te njegovo proširenje u \(d\)-dimenzionalnom prostoru.

Abstract (english)

Poncelet closure theorem, which is a theorem about closing property for a sequence of polygons inscribed in one and circumscribed about the other given conic, is one of the most explored and most important theorems of projective geometry. Besides Poncelet theorem, there are more theorems of this type, such as Steiner theorem, "Zigzag" theorem and Emch theorem, which is in a sense the strongest one among these four. At the beginning, we present a generalized closing property on a circle \(k\) in the plane and later on a corresponding property is formulated for spheres and a given circle \(k\) in \(d\)-dimensional Euclidean space. It is proved that any family of spheres \(\mathcal{F}\) , that satisfies certain nonsingularity conditions and a simple analytical relation, possesses the closing property on a circle. In that case, the centres of all spheres from \(\mathcal{F}\) belong to a certain quadric \(\Gamma\). The geometrical meaning of this condition is that spheres from \(\mathcal{F}\) are tangential to two spheres \(S_0\) and \(S_1\) and orthogonal to a given sphere \(Q\). It is shown that four classical theorems of Poncelet type are corollaries of the general closing property for spheres. In the last section, we focus on the elementary proof of Emch’s theorem in the plane and its extension to the Euclidean space \(\mathbb{R}^d\).

• Keywords: Ponceletov teorem; Steinerov teorem; zigzag teorem; Emchov teorem
• Keywords (english): Poncelet closure theorem; Steiner theorem; zigzag theorem; Emch theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:587634
• Study programme: Title: Finance and Business Mathematics; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra matematike (magistar/magistra matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2017-06-09 09:21:33