Perspektivno preslikavanje kvadrata

master's thesis

• Author: Hana Kovačić
• Mentor: Juraj Šiftar (mentor)
• Committee member: Juraj Šiftar (predsjednik povjerenstva)
• Committee member: Boris Širola (član povjerenstva)
• Committee member: Sonja Štimac (član povjerenstva)
• Committee member: Franka Miriam Bruckler (član povjerenstva)
• Granter: University of Zagreb; Faculty of Science; (Department of Mathematics); Zagreb
• Defense date and country: 2019-07-15, Croatia
• Scientific / art field, discipline and subdiscipline: NATURAL SCIENCES; Mathematics

Abstract

Teorem koji iskazuje naizgled jednostavnu geometrijsku činjenicu da je svaki četverokut perspektivna slika nekog kvadrata može se smatrati odavno poznatim, jer su ga barem intuitivno otkrivali već neki slikari u doba renesanse. Korektna formulacija i različiti dokazi, koji obuhvaćaju tu tvrdnju u punoj općenitosti, pojavljuju se međutim dosta rijetko i to tek u novijoj literaturi. U ovom radu taj teorem tumači se polazeći najprije od klasičnih primjera perspektive odnosno centralne projekcije, na temelju pristupa kakav je sustavno izložio znameniti matematičar Brook Taylor u svojoj knjizi Novi principi linearne perspektive iz 1719. godine. Zatim se izlaže stereometrijski dokaz A. Emcha te se promatraju kvadratne konike, kao presjeci dvostrukog kvadratnog stošca ravninama. Nadalje, teorem o perspektivnoj slici kvadrata povezuje se s temeljnim teoremom projektivne geometrije, čime se dobivaju neki zanimljivi rezultati o dekompoziciji projektivnih kolineacija. Takve dekompozicije imaju važno mjesto u suvremenim primjenama kao što su računalni vid, animacija i analiza video materijala. U završnom dijelu, na jednom primjeru prikazana je analiza stvarnih relativnih dimenzija likova na fotografiji primjenom zakonitosti perspektive, povezana s teoremom o preslikavanju kvadrata i s interpretacijom na Taylorov način.

Abstract (english)

A theorem, which states a seemingly simple geometric fact that every quadrangle is a perspective image of a square, may be considered as known for a long time. Namely, even some renaissance painters have discovered it, albeit on an intuitive level. However, a correct formulation together with various proofs that cover this statement in full generality are not so common and can be found only in recent literature. In the first part, this thesis approaches the theorem by analysing classic examples of a perspectivity or, more specifically, central projections. This method was systematically treated by famous mathematician Brook Taylor in his book New Principles of Linear Perspective from 1719. Next, a stereometric proof of A. Emch is provided and square conics are observed, as sections of a double square cone with planes. Furthermore, the theorem on perspective images of a square is associated with the fundamental theorem of projective geometry, which gives interesting results on decomposition of projective collineations. These types of decompositions are of great importance in modern applications such as computer vision, animation and video analysis. In the final part, an analysis of real relative dimensions of objects on a photograph is shown by applying perspectivity laws. This analysis is brought into relation with the theorem on mapping of a square and Taylor’s interpretation.

• Keywords: Brook Taylor; stereometrijski dokaz; kvadratne konike; teorem o perspektivnoj slici kvadrata; dekompozicija projektivnih kolineacija
• Keywords (english): Brook Taylor; stereometric proof; square conics; theorem on perspective images of a square; decomposition of projective collineations
• Language: croatian
• URN:NBN: urn:nbn:hr:217:602536
• 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: 2019-09-30 10:42:25