Topološka analiza 3D modelov

Treven, Marjana (author), Gabrovšek, Boštjan (mentor)

V magistrskem delu je predstavljena homologija poljubnega telesa ali ploskve. Homologija nam pomaga prepoznati, šteti in meriti različne vrste ”lukenj” v prostoru, kot so vrzeli, zanke in votline. V delu se posebej osredotočamo na teoretično ozadje in samo definicijo homologije, ki jo definiramo preko vektorskih prostorov. Sledi nekaj primerov izračunov homologije ter predstavitev programske kode, ki za poljubno triangulirano telo ali ploskev izračuna dimenzije homoloških vektorskih prostorov. V zadnjem delu magistrske naloge so predstavljene aplikacije, kjer se uporabljajo homološke metode. S pomočjo homologije lahko na primer odkrivamo luknje v brezžičnih omrežjih, pomaga pa nam tudi pri odkrivanju rakastih tkiv.

• Language: Slovenian
• Keywords: vektorski prostori, simpleksi, simplicialni kompleksi, robne preslikave, verižni kompleksi, homološki vektorski prostori
• Work type: Master's thesis/paper
• Organization: FS - Faculty of Mechanical Engineering
• Place of publishing: Ljubljana
• Publisher: [M. Treven]
• Year: 2023
• Number of pages: XVII, 52 str.
• PID: 20.500.12556/RUL-152433
• UDC: 512.642:515.142.33(043.2)
• COBISS.SI-ID: 183202051
• Publication date in RUL: 24.11.2023

TREVEN, Marjana, 2023, Topološka analiza 3D modelov [online]. Master’s thesis. Ljubljana : M. Treven] . [Accessed 14 April 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=152433

Secondary language

• Language: English
• Title: Topological analysis of 3D models

Abstract

In the master's thesis we present the homology of a body or surface. Homology helps us recognize, count and measure different types of "holes" in space, such as gaps, loops or cavities. The main focus of the work is on the theoretical background and on the definition of homology which is defined through vector spaces. We presented a few examples of homology calculations and wrote a program code that calculates the dimension of homology vector spaces for any triangulated body or surface. The final part of the thesis presents applications where homological methods are used. For example, with the help of homology, we can detect holes in wireless networks, and it also aids us in the detection of cancerous tissues.

• Keywords: vector spaces, simplices, simplicial complexes, boundary maps, chain complexes, homology vector spaces