Diofantski vidiki Steinerjevega porizma

magistrsko delo

Nejc Barovič (Author), Bojan Hvala (Mentor)

Abstract

Na začetku magistrskega dela je predstavljena preslikava inverzija na krožnici z njenimi najpomembnejšimi lastnostmi. Vpeljana sta tudi pojma notranje in zunanje podobnostno središče dveh krožnic ter natančno je razložena povezava s podobnostno preslikavo. Drugo poglavje je namenjeno Steinerjevemu porizmu. Prikazani so trije načini konstrukcije prve Steinerjeve krožnice, konstrukcija Steinerjeve verige z uporabo inverzije in brez njene uporabe ter dokaz Steinerjevega porizma. Nato sta izpeljani Steinerjeva formula in posplošena Steinerjeva formula. S formulama dobimo potrebne in zadostne pogoje za sklenitev Steinerjeve verige v enem oziroma v m obhodih. V četrtem poglavju so obravnavane diofantske Steinerjeve trojke. Dokazano je, da je sklenjena Steinerjeva veriga v enem obhodu pri diofantskih Steinerjevih trojkah lahko le dolžine 3, 4 ali 6. Za vsakega od teh primerov je izpeljana in rešena diofantska enačba. Dokazano je tudi, da odprava omejitve, da se mora veriga skleniti že po prvem obhodu, ne prinese nobenih novih rešitev: če se veriga sklene po m obhodih, se je sklenila že po prvem obhodu. Zadnji del je posvečen polmerom krožnic v verigi. Ti se seveda spreminjajo glede na začetne podatke. Dokazano je, da lahko vhodne parametre izberemo tako, da so tudi vsi polmeri krožnic v verigi naravni.

Keywords

inverzija;Steinerjeva veriga;Steinerjev porizem;Steinerjeva formula;diofantske Steinerjeve trojke;polmeri Steinerjevih krožnic;magistrska dela;

Data

Language:	Slovenian
Year of publishing:	2014
Typology:	2.09 - Master's Thesis
Organization:	UM FNM - Faculty of Natural Sciences and Mathematics
Publisher:	[N. Barovič]
UDC:	514.112:511(043.2)
COBISS:	20631560
Views:	1447
Downloads:	217
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	The Diophantine Aspects in Steiner's Porism
Secondary abstract:	In this master's thesis I discuss the Diophantine aspects in Steiner's porism. In the first part I discuss the transformation of an inversion on a circle with its most important characteristics. I have also introduced two notions, i.e. internal and external similitude centre of two circles. The connection with the homothetic transformation is also precisely explained. The second chapter discusses Steiner's porism. The three ways of constructing the Steiner's circle are explained, i.e. the construction of Steiner chain with or without the use of the inversion, and the proof of Steiner's porism. Later-on the Steiner formula and the simplified Steiner formula are derived. By using the mentioned formulas we obtain the necessary and sufficient conditions to conclude the Steiner chain in one or m warps. The fourth chapter discusses the Diophantine Steiner triples. It is proven that a concluded Steiner chain in one round when discussing the Diophantine Steiner triples can only measure 3, 4 or 6 in length. For each of the previously mentioned examples a Diophantine equation is derived and solved. It is also proven that the elimination of a restriction, i.e. the need of a chain to be concluded after the first warp, does not bring forth no new solutions. Therefore, if a chain is concluded after m warps, it has already been concluded after the first one. The final part of the thesis is dedicated to radii of circles in a chain. These change according to the initial data. It is proven that we can choose the incoming parameters in such a way in which all radii of circles in a chain are positive integer.
Secondary keywords:	inversion;Steinerʼs chain;Steinerʼs porism;Steinerʼs formula;diophantine Steinerʼs triples;radius of the steinerʼs circles;master theses;
URN:	URN:SI:UM:
Type (COBISS):	Master's thesis
Thesis comment:	Univ. v Mariboru, Fak. za naravoslovje in matematiko, Oddelek za matematiko in računalništvo
Pages:	XIII, 94 f.
ID:	8729532