Evesov teorem o projektivnim invarijantama

master's thesis

• Author: Ivana Garić
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Ozren Perše (član povjerenstva)
• Committee member: Marcela Hanzer (član povjerenstva)
• Committee member: Zlatko Drmač (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2018-09-24, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Pojam h-izraza uveo je Howard Eves kao poopćenje različitih umnožaka omjera usmjerenih dužina kakvi se pojavljuju u važnim rezultatima euklidske geometrije poput Cevinog i Menelajevog teorema. S druge strane, h-izraz je poopćenje dvoomjera, kao osnovne numeričke invarijante projektivne geometrije. Središnja tema ovog rada je Evesov teorem koji govori da je h-izraz projektivna invarijanta. Taj teorem pruža mogućnost raznovrsnih primjena, kako u dokazivanju geometrijskih teorema tako i u praktičnim primjerima, poput fotogrametrije gdje se mjerenjima na dvodimenzionalnoj slici rekonstruiraju odnosi veličina u trodimenzionalnom originalu. Nadalje, prikazana su osnovna svojstva i primjene jedne geometrijske invarijante koja je po obliku bliska Evesovom h-izrazu, ali se odnosi samo na izomorfizme projektivnih konfiguracija, dakle nisu nužno inducirane projektivnim transformacijama ravnine. Ključnu ulogu ovdje imaju tzv. Nehringovi nizovi točaka i pravaca. Oni se mogu pridružiti različitim konfiguracijama, a varijacijama izbora početne točke pripadni h-izraz ostaje konstantan te je stoga invarijanta izomorfizma.

Abstract (english)

The concept of the h-expression was introduced by Howard Eves as a generalization of various products of ratios of directed lengths that appear in important results of Euclidean geometry like Ceva's and Menelaus's theorems. On the other hand, h-expression is a generalization of the cross-ratio as a basic numeric invariant of projective geometry. The central topic of this paper is Eves's theorem which states that h-expression is a projective invariant. This theorem provides the possibility of a variety of applications, both in proving geometric theorems as well as in practical examples, such as a photogrammetry where the measurements in a two-dimensional image are used to reconstruct the relationships between sizes in the three-dimensional original. Furthermore, the basic properties and applications of another geometric invariant are presented. This geometric invariant is closely related to Eves's h-expression, but it refers only to the isomorphisms of projective configurations. Therefore, they are not necessarily induced by projective plane transformations. Nehring's sequences of points and lines are crucial in this approach. They can be associated with different configurations, and the corresponding h-expression remains constant with regard to variation of the choice of the starting point. Hence, it is an isomorphism invariant.

• Keywords: h-izraz; euklidska geometrija; Evesov teorem; fotogrametrija; Nehringovi nizovi točaka i pravaca
• Keywords (english): h-expression; Euclidean geometry; Eves's theorem; photogrammetry; Nehring's sequences of points and lines
• Language: croatian
• URN:NBN: urn:nbn:hr:217:374495
• 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: 2019-04-23 10:42:01