Steinerjev porizem in njegova prostorska analogija

magistrsko delo

Nejc Kastelic (Author), Aleš Vavpetič (Mentor)

Abstract

Steinerjevo verigo sestavlja zaporedje krožnic tangentnih na dani dve nesekajoči se krožnici, zaporedna člena zaporedja pa sta med sabo tangentna. Inverzija na krožnico, o kateri dokažemo nekaj lastnosti, nam problem prevede v iskanje Steinerjeve verige med dvema koncentričnima krožnicama. Uspeli smo poiskati tudi formulo, preko katere lahko hitro preverimo, ali Steinerjeva veriga obstaja ali ne in iz koliko krožnic je sestavljena. Nastavili smo tudi pot iskanja rešitve preko Möbiousovih transformacij. Analogija v prostoru krožnice nadomesti s sferami, Steinerjevo verigo pa nadomesti ustrezna Steinerjeva družina sfer, katere središča v koncentričnem primeru so oglišča ustreznih Platonskih teles. Rešitve proučujemo z obravnavo prostorskih kotov. Tudi tukaj smo našli formulo, ki pove za kateri sferi je sploh možno poiskati take družine sfer.

Keywords

matematika;inverzija čez krožnico;Steinerjev porizem;Steinerjeva veriga;potenca točke;inverzija v prostoru;diederski kot;prostorski kot;Platonsko telo;Eulerjeva poliederska formula;mathematics;circle inversion;Steinerʼs porism;Steine chain;pointʼs circle power;sphere inversion;dihedral angle;solid angle;Platonic polyhedra;Eulerʼs polyhedrom formula;

Data

Language:	Slovenian
Year of publishing:	2019
Typology:	2.09 - Master's Thesis
Organization:	UL FMF - Faculty of Mathematics and Physics
Publisher:	[N. Kastelic]
UDC:	514
COBISS:	18632793
Views:	1149
Downloads:	243
Average score:	0 (0 votes)
Metadata:

Other data

Secondary language:	English
Secondary title:	Steiner chain and its spatial analogy
Secondary abstract:	The Steiner chain consists of a set of circles, which are tangent to two given non-intersecting circles and each circle in the chain is tangent to the previous and next circle in the chain. Inversion in a circle, for which we have proven some properties, translates the problem into the search of a Steiner chain between two concentric circles. We have also succeeded in finding a formula that makes it easier to check whether the Steiner chain exists and how many circles it consists of. Through analogy in space, circles are replaced by spheres, whereas the Steiner chain is replaced by the Steiner family of spheres whose centers in concentric cases are the angles of corresponding Platonic solids. Solutions are studied through the treatment of solid angles. Here too, we have found a formula which tells us which spheres it is possible to find the Steiner family of spheres for.
Secondary keywords:	circle inversion;Steiner's Porism;Steiner chain;point's circle power;sphere inversion;dihedral angle;solid angle;Platonic polyhedra;Euler's polyhedron formula;
Type (COBISS):	Master's thesis/paper
Study programme:	0
Embargo end date (OpenAIRE):	1970-01-01
Thesis comment:	Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Pedagoška matematika
Pages:	IX, 46 str.
ID:	11153058