magistrsko delo
Aljaž Bogataj (Author), Marjetka Krajnc (Mentor)

Abstract

Pomembno orodje na področju računalniško podprtega geometrijskega oblikovanja so krivulje s pitagorejskim hodografom (PH krivulje), saj so gladke, računsko učinkovite in omogočajo enostaven analitičen izračun nekaterih osnovnih lastnosti parametrično podanih krivulj. Te krivulje lahko obravnavamo v evklidskih in Minkowskijevih prostorih poljubnih dimenzij. Kljub temu da temeljijo na enaki ideji, se njihova karakterizacija precej razlikuje. Na primer, PH krivulje v evklidski ravnini ${\mathbb R}^2$ najlažje izrazimo s kompleksnimi števili, medtem ko PH krivulje v ${\mathbb R}^3$ izrazimo s kvaternioni. V ta namen v delu predstavimo karakterizacijo poljubnih PH krivulj v enotnem ogrodju Cliffordove algebre. Krivulje definiramo s PH reprezentacijsko preslikavo in ustreznost definicije preverimo s konstrukcijo že znanih PH krivulj. Posebno pozornost namenimo redko obravnavanim PH krivuljam v štiridimenzionalnem prostoru Minkowskega ${\mathbb R}^{3,1}$. Uporabnost enotnega ogrodja demonstriramo s primeri konstrukcije različnih vrst PH krivulj z uporabo Bézierjevih krivulj, kjer se razlike v konstrukciji pojavijo le pri izbiri praslike PH reprezentacijske preslikave ter izbiri stopnje Bézierjeve krivulje.

Keywords

krivulje s pitagorejskim hodografom;PH reprezentacijska preslikava;Cliffordova algebra;prostor Minkowskega;

Data

Language: Slovenian
Year of publishing:
Typology: 2.09 - Master's Thesis
Organization: UL FMF - Faculty of Mathematics and Physics
Publisher: [A. Bogataj]
UDC: 519.6
COBISS: 246578691 Link will open in a new window
Views: 74
Downloads: 19
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Other data

Secondary language: English
Secondary title: Construction of curves with pythagorean properties using Clifford algebra
Secondary abstract: An important tool in the field of Computer Aided Geometric Design are Pythagorean hodograph (PH) curves, as they are smooth, computationally efficient, and allow the straightforward analytical computation of some of their properties. Moreover, these curves can be considered in both Euclidean and Minkowski spaces of arbitrary dimension. Although their construction is based on the same underlying idea, their characterization varies significantly. For example, PH curves in the Euclidean plane ${\mathbb R}^2$ are most conveniently expressed using complex numbers, while PH curves in ${\mathbb R}^3$ are typically expressed with quaternions. To address these differences, we present a unified characterization of general PH curves within the framework of Clifford algebra. We define these curves with the PH representation map and verify the validity of the definition through the construction of already known PH curves. Special attention is given to the less frequently studied case of PH curves in the four-dimensional Minkowski space ${\mathbb R}^{3,1}$. The practical utility of the unified framework is demonstrated through numerical construction of various types of PH curves using Bézier curves. It is shown that differences in the construction are solely in the choice of the preimage of the PH representation map and in the degree of the Bézier curve.
Secondary keywords: Pythagorean hodograph curves;PH representation map;Clifford algebra;Minkowski space;
Type (COBISS): Master's thesis/paper
Study programme: 0
Thesis comment: Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Pedagoška matematika
Pages: VI, 53 str.
ID: 27174938