Euler-Rodriguesova ogrodja prostorskih krivulj s pitagorejskim hodografom
delo diplomskega seminarja
Abstract
V diplomskem delu se bomo posvetili metodam, ki so uporabne v računalniško podprtem geometrijskem oblikovanju. Natančneje si bomo pogledali gibanje togega telesa v evklidskem tridimenzionalnem prostoru, ki ga v splošnem določimo z opisom lokacije centra togega telesa in orientacije v odvisnosti od časa. Pri opisu bomo spoznali prostorske krivulje s pitagorejskim hodografom, ki imajo v računalniško podprtem geometrijskem oblikovanju pomembno vlogo, saj njihova polinomska formulacija omogoča racionalen zapis enotskega tangentnega vektorja, ortonormiranega ogrodja, dolžine krivulje itd. Racionalne oblike so pomembne, ker omogočajo učinkovite in natančne izračune. Tako bomo spoznali racionalno Euler-Rodriguesovo ogrodje, ki je naravno definirano na kvaternionski reprezentaciji prostorskih krivulj s pitagorejskim hodografom. Videli bomo, da to ogrodje v splošnem izvaja nepotrebne rotacije, ki v računalniškem modeliranju povzročajo popačenje slik. Ogrodja, ki takih rotacij ne izvajajo, imenujemo rotacijsko minimizirajoča ogrodja. Čeprav v splošnem Euler-Rodriguesovo ogrodje ni rotacijsko minimizirajoče, bomo spoznali pogoje na prostorske krivulje s pitagorejskim hodografom, kjer to velja.
Keywords
matematika;Euler-Rodriguesovo ogrodje;krivulja s pitagorejskim hodografom;racionalno ortonormirano ogrodje;rotacijsko minimizirajoče ogrodje;kvaternioni;
Data
| Language: | Slovenian |
|---|---|
| Year of publishing: | 2021 |
| Typology: | 2.11 - Undergraduate Thesis |
| Organization: | UL FMF - Faculty of Mathematics and Physics |
| Publisher: | [G. Mrzelj] |
| UDC: | 519.6 |
| COBISS: |
73961475
|
| Views: | 1306 |
| Downloads: | 66 |
| Average score: | 0 (0 votes) |
| Metadata: |
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Other data
| Secondary language: | English |
|---|---|
| Secondary title: | Euler-Rodrigues frames on spatial Pythagorean-hodograph curves |
| Secondary abstract: | In this thesis we will focus on methods that are useful in computer aided geometric design. We will take a closer look at the motion of a rigid body in Euclidean three-dimensional space, which is generally determined by describing the location of the center of the rigid body and orientation as a function of time. In the description of these curves, we will get to know spatial Pythagorean-hodograph curves, which play an important role in computer aided geometric design, as their polynomial formulation allows a rational representation of the unit tangent vector, orthonormal frame, arc length, etc. Rational forms are important since they allow efficient and accurate calculations. Thus, we will get to know the rational Euler-Rodrigues frame, which is naturally defined on the quaternion representation of spatial Pythagorean-hodograph curves. We will see that this frame generally performs unnecessary rotations that cause image distortion in computer modeling. Frames that do not perform such rotations are called rotation minimizing frames. Although in general the Euler-Rodrigues frame is not rotation minimizing, we will derive the conditions on the spatial Pythagorean-hodograph curves where this applies. |
| Secondary keywords: | mathematics;Euler-Rodrigues frame;Pythagorean-hodograph curve;rational orthonormal frame;rotation-minimizing frame;quaternions; |
| Type (COBISS): | Final seminar paper |
| Study programme: | 0 |
| Embargo end date (OpenAIRE): | 1970-01-01 |
| Thesis comment: | Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Matematika - 1. stopnja |
| Pages: | 36 str. |
| ID: | 13243602 |
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