Steinerjev porizem in njegova prostorska analogija : magistrsko delo

Kastelic, Nejc (author), Vavpetič, Aleš (mentor)

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.

• Language: Slovenian
• Keywords: inverzija čez krožnico, Steinerjev porizem, Steinerjeva veriga, potenca točke, inverzija v prostoru, diederski kot, prostorski kot, Platonsko telo, Eulerjeva poliederska formula
• Work type: Master's thesis/paper
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2019
• PID: 20.500.12556/RUL-107930
• UDC: 514
• COBISS.SI-ID: 18632793
• Publication date in RUL: 07.06.2019

Secondary language

• Language: English
• Title: Steiner chain and its spatial analogy

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.

• Keywords: circle inversion, Steiner's Porism, Steiner chain, point's circle power, sphere inversion, dihedral angle, solid angle, Platonic polyhedra, Euler's polyhedron formula