Krivulje konstantne širine

master's thesis

• Author: Mateja Miklenić
• Mentor: Tomislav Pejković (mentor)
• Committee member: Tomislav Pejković (predsjednik povjerenstva)
• Committee member: Tina Bosner (član povjerenstva)
• Committee member: Pavle Pandžić (član povjerenstva)
• Committee member: Maja Starčević (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-25, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Svakoj zatvorenoj krivulji možemo odrediti širinu u bilo kojem smjeru. Krivulju kojoj je širina jednaka u svakom smjeru nazivamo krivuljom konstantne širine. Trivijalni primjer takve krivulje je kružnica. Osim kružnice, postoji beskonačno mnogo krivulja konstantne širine. Najjednostavniji primjer takve krivulje je krivolinijski trokut, kojeg je otkrio njemački znanstvenik i inženjer Franz Reuleaux, prema kojem ga nazivamo Reuleauxovim trokutom. U prvom dijelu rada dani su i drugi načini konstrukcije krivulja konstantne širine. Krivulje konstantne širine imaju mnoga zanimljiva svojstva, od kojih su neka navedena i dokazana u drugom dijelu ovog rada. Primjerice, Barbierov teorem kaže da krivulja konstantne širine h i kružnica promjera h imaju jednaku duljinu. Iskazane su i ilustrirane i dvije tvrdnje o duljini i površini obuhvaćenoj takvim krivuljama, a dokazan je i Aproksimacijski teorem za krivulje konstantne širine. Krivulja konstantne širine čini rub lika kojeg nazivamo likom konstantne širine. Prostorni analogon lika konstantne širine je tijelo konstantne širine. Prema načinu na koji ih dobivamo, dijelimo ih na rotacijska i nerotacijska tijela. Otkriće dva tipa nerotacijskih tijela konstantne širine pripisuje se švicarskom matematičaru Ernstu Meissneru. Kratko izlaganje o tijelima konstantne širine dano je u završnom dijelu ovog rada.

Abstract (english)

For each closed curve we can determine the width in any direction. If the width of a curve is the same in all directions, then it is called a curve of constant width. A common example of such a curve is a circle. Apart from the circle, there are infinitely many other curves of constant width. The simplest example of such curves is the curvilinear triangle, also called the Reuleaux triangle because it was discovered by German scientist and engineer Franz Reuleaux. In the first part of this graduate thesis we describe different ways of constructing curves of constant width. Curves of constant width have many interesting properties, and some of them are mentioned and proven in this work. For example, Barbier’s theorem says that a curve of constant width h and a circle of diameter h have the same length. This and other statements on length and the area enclosed by such curves are given and illustrated. The so called Approximation Theorem for curves of constant width is also proved. A curve of constant width is the boundary of the figure that we call a figure of constant width. The three-dimensional analogy of the figure of constant width is the body of constant width. According to the way they are obtained, they are divided into rotational and non-rotational bodies. Two types of non-rotational bodies of constant width were discovered by Swiss mathematician Ernst Meissner. A short exposition on bodies of constant width makes the last part of this thesis.

• Keywords: krivulja konstantne širine; Reuleauxov trokut; Barbierov teorem; Aproksimacijski teorem za krivulje konstantne širine; tijelo konstantne širine; rotacijska tijela; nerotacijska tijela; Meissner
• Keywords (english): curve of constant width; Reuleaux triangle; Barbier’s theorem; Approximation Theorem for curves of constant width; figure of constant width; rotational bodies; non-rotational bodies; Meissner
• Language: croatian
• URN:NBN: urn:nbn:hr:217:574923
• 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: Open access
• Created on: 2019-06-07 12:59:37