Fermat-Steinerov problem

master's thesis

• Author: Monika Cikojević
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Filip Najman (član povjerenstva)
• Committee member: Sanja Varošanec (član povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2015-09-24, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Fermat-Steinerov problem je klasičan matematički problem kojim se niz matematičara bavio još od 17. stoljeća. Problem se sastoji u određivanju takve točke u ravnini zadanog trokuta da zbroj njezinih udaljenosti od vrhova trokuta bude minimalan. Zanimljivo je da se rješenje može naći potpuno različitim pristupima, od čisto geometrijskih, preko diferencijalnog računa, do praktičnog mehaničkog rješenja u kojem se koristi princip minimalne energije. Motivacija i za osnovni i za težinski problem može se pronaći u primjenama za različite zadatke optimizacije, npr. tri sela ili tri tvornice. Pokazuje se da se Fermat-Steinerova točka, koja predstavlja rješenje problema, može geometrijski karakterizirati i na druge zanimljive načine. Nadalje, razmatra se inverzni problem težinskog Fermat-Steinerova problema u kojem se za zadanu točku unutar trokuta ispituje postojanje težina takvih da je upravo ta točka rješenje. Iznijeli smo više različitih dokaza kako bismo prikazali povijesni razvitak pristupa problemu i njegovim generalizacijama. Time smo ujedno željeli pokazati ljepotu i raznovrsnost matematičkog promišljanja.

Abstract (english)

The Fermat-Steiner problem is a classical mathematical problem which numerous mathematicians have dealt with since the 17th century. The gist of the problem is determining such a point in the plane of a given triangle, so that the sum of the distances between that point and the vertices of the triangle is minimal. It is interesting that the solution can be found by completely different approaches, from purely geometric, by using calculus and by using practical mechanical method in which minimum energy principle is used. Motivation for the basic and weighted problem can be found in its usage for different optimization tasks, e.g. three villages or three factories. The Fermat-Steiner point, which is the solution of the problem, can be geometrically characterized in various interesting ways. Furthermore, the inverse problem of the weighted Fermat-Steiner problem is considered, in which it is investigated whether there exist such weights that a given point is the solution of the weighted problem. We have put forward various different proofs in order to show the historical development of the approach to the problem, and its generalizations. In that way we also wanted to show the beauty and diversity of mathematical thought.

• Keywords: Fermat-Steinerova točka; zadaci optimizacije; težinski Fermat-Steinerov problem
• Keywords (english): Fermat-Steiner point; optimization tasks; weighted Fermat-Steiner problem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:407345
• Study programme: Title: Mathematics Education; specializations in: Mathematics Education; Course: Mathematics Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike (magistar/magistra edukacije matematike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Access restricted to students and staff of home institution
• Created on: 2019-01-07 10:21:24