Ekstremalna svojstva paralelograma upisanih u elipsu

master's thesis

• Author: Marina Križan
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Tomislav Berić (član povjerenstva)
• Committee member: Dijana Ilišević (član povjerenstva)
• Committee member: Miljenko Marušić (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-27, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Među mnogobrojnim svojstvima paralelograma upisanih u elipsu, u ovom radu su istaknuta dva koja su povezana s maksimalnim opsegom i maksimalnom površinom paralelograma. Koristeći neke elementarne činjenice o krivuljama drugog reda iz afine i projektivne geometrije, dokazan je Glavni teorem koji kaže da su paralelogrami maksimalnog opsega upisani u elipsu karakterizirani svojstvom da su tangente elipse u susjednim vrhovima paralelograma međusobno okomite, te da je svaka točka elipse vrh točno jednog takvog paralelograma. U radu je opisano i fizikalno tumačenje problema određivanja svih paralelograma maksimalnog opsega upisanih u zadanu elipsu koji je Jean-Marc Richard povezao s problemom iz balistike o određivanju „sigurnog područja“ za familiju putanja hitaca, odnosno satelita. Nadalje, izloženi su još neki alternativni dokazi i poopćenja Glavnog teorema koje su naveli A. Connes i D. Zagier. Opisani su i rezultati N. Anghela čiji je glavni cilj bio poopćenje u smjeru karakterizacije klase krivulja kojima je ortoptička krivulja kružnica. Pozivanjem na nekoliko stavaka iz Bergerove „Geometrije“ zaokružen je pregled prethodno izloženih rezultata te je istaknuta veza s klasičnim Ponceletovim teoremom o poligonima upisanima jednoj, a opisanima drugoj zadanoj krivulji 2. reda. Na kraju rada je izloženo rješenje problema u vezi određivanja paralelograma maksimalne površine upisanih u elipsu koristeći činjenicu da se elipsa može prikazati kao slika kružnice u perspektivnoj afinosti te da je omjer površina invarijanta svakog afinog preslikavanja.

Abstract (english)

Among the many properties of parallelograms inscribed in the ellipse, here are presented two of these which are related to the maximal perimeter and maximal area of the parallelogram. Using some elementary facts about curves of the second order from affine and projective geometry, the Main Theorem is proven, asserting that parallelograms of maximal perimeter, inscribed in an ellipse, are characterized by the fact that the tangents of ellipse at adjacent vertices of the parallelogram are mutually perpendicular. Moreover, each point of the ellipse is a vertex of exactly one such parallelogram. A physical interpretation of the problem of determining all parallelograms of maximal perimeter inscribed in a given ellipse is also described in this paper. Jean-Marc Richard associated this problem to a problem from ballistics, namely determining the “safe domain" for the family of trajectories of projectiles or satellites. Furthermore, some alternative proofs and generalizations of the Main Theorem are presented, as stated by A. Connes and D. Zagier. Some related results of N. Anghel are described as well, his main goal being a characterization of a larger class of curves whose orthoptic curve is a circle. By referring to several items from M. Berger’s ”Geometry”, a review of previously presented results is completed, indicating also a link to the classic Poncelet theorem on polygons inscribed in one given conic and circumscribed about another given conic. In the final chapter, a solution to the problem of determining parallelograms with maximal area inscribed in an ellipse is presented, using a representation of the ellipse as the image of a circle in perspective affinity and the affine invariance of the ratio of areas.

• Keywords: paralelogram; elipsa; Richard; Connes; Zagier; Anghel; Berger; Ponceletov teorem
• Keywords (english): parallelogram; ellipse; Richard; Connes; Zagier; Anghel; Berger; Poncelet theorem
• Language: croatian
• URN:NBN: urn:nbn:hr:217:193324
• 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: 2019-05-27 11:46:31