Spiralna sličnost

master's thesis

• Title (english): Spiral similarity
• Author: Dijana Jurić
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Mea Bombardelli (član povjerenstva)
• Committee member: Mladen Jurak (član povjerenstva)
• Committee member: Zlatko Drmač (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2021-03-05, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Spiralna sličnost je preslikavanje euklidske ravnine zadano kao kompozicija homotetije i rotacije sa zajedničkim centrom. U ovom radu prikazani su osnovni teoremi o spiralnoj sličnosti te neke njezine primjene u rješavanju složenijih zadataka iz područja planimetrije. U uvodnom poglavlju dan je pregled poznatih preslikavanja sličnosti i istaknuto je mjesto spiralne sličnosti među njima. Zatim se uočava da je spiralnu sličnost praktično prikazati kao geometrijsku interpretaciju umnoška dvaju kompleksnih brojeva te se dokazuje egzistencija i jedinstvenost spiralne sličnosti koja za zadane dvije dužine preslikava jednu dužinu u drugu. U daljnjim teoremima opisuje se određivanje centra spiralne sličnosti za različite slučajeve te se uočava da spiralne sličnosti uvijek dolaze u paru. Definira se pojam simedijane u trokutu i pokazuje se povezanost simedijane s položajem centra spiralne sličnosti. U svrhu rješavanja zadataka u kojima se primjenjuje spiralna sličnost navedeni su još neki pojmovi odnosno dokazani pojedini rezultati, uglavnom povezani s geometrijom kružnice (Apolonijeva kružnica, inverzija, Pascalov teorem). Završno, najopsežnije poglavlje sastoji se od rješenja nekoliko zadataka s Međunarodne matematičke olimpijade (IMO) te s matematičkih natjecanja u Japanu, Južnoj Koreji i Iranu. U svim zadatcima primjenjuje se spiralna sličnost. To nije uvijek i najprikladniji način rješavanja, ali su primjeri pogodni za uočavanje uloge spiralne sličnosti.

Abstract (english)

Spiral similarity is the mapping of the Euclidean plane given as a composition of homothety and rotation with a common center. This paper presents the basic theorems on spiral similarity and some of its applications in solving complex problems in the field of planimetry. The introductory chapter gives an overview of the known similarity mappings and highlights the place of the spiral similarity among them. It is noticed that it is useful to present the spiral similarity as a geometric interpretation of the product of two complex numbers, and the existence and uniqueness of the spiral similarity is proven, which for a given pair of line segments maps one onto another. Further theorems describe the determination of the center of spiral similarity in different cases and it is observed that spiral similarities always occur in pairs. The notion of a symmedian in a triangle is defined and its connection with the position of the center of the spiral similarity is explained. In order to solve problems in which spiral similarity is applied, some other concepts and results are set forth, mainly related to the geometry of a circle, such as the Apollonius circle, inversion and Pascal’s theorem. The final and the most comprehensive chapter consists of solutions to several problems from the International Mathematical Olympiad (IMO) and from mathematical competitions in Japan, South Korea and Iran. In each solution, spiral similarity is applied. This approach is not always the most practical one, but the examples are suitable for observing the role of spiral similarity.

• Keywords: euklidska ravnina; planimetrija; simedijana; geometrija kružnice
• Keywords (english): Euclidean plane; planimetry; symmedian; geometry of a circle
• Language: croatian
• URN:NBN: urn:nbn:hr:217:796819
• Study programme: Title: Mathematics and Computer Science Education; specializations in: Mathematics and Computer Science Education; Course: Mathematics and Computer Science Education; Study programme type: university; Study level: graduate; Academic / professional title: magistar/magistra edukacije matematike i informatike (magistar/magistra edukacije matematike i informatike)
• Type of resource: Text
• File origin: Born digital
• Access conditions: Open access
• Created on: 2021-03-24 13:20:34