Stožnice in kvadriki v racionalni Bézierjevi obliki
magistrsko delo
Abstract
Delo obravnava stožnice, podane kot racionalne Bézierjeve krivulje stopnje 2. V delu opišemo in ponazorimo algebraične in geometrijske lastnosti stožnic, podanih v taki obliki. Izpeljemo postopek za prehod iz racionalne Bézierjeve oblike stožnice v implicitno obliko in obratno. Na podlagi števila singularnosti stožnice, podane v racionalni Bézierjevi obliki, določimo njen tip. Podrobneje se posvetimo krožnemu loku oz. krožnici in pokažemo, da za predstavitev celotne krožnice potrebujemo racionalno Bézierjevo krivuljo stopnje vsaj 5. Nato se posvetimo še predstavitvi kvadrikov z racionalnimi Bézierjevimi ploskvami. Pri tem se osredotočimo predvsem na sfero. S pomočjo stereografske projekcije predstavimo oktant sfere kot racionalno trikotno Bézierjevo krpo, šestino sfere pa kot racionalno Bézierjevo ploskev iz tenzorskega produkta.
Keywords
Bézierjeve krivulje;stožnice;baricentrične koordinate;Bézierjeve ploskve;kvadriki;stereografska projekcija;
Data
| Language: | Slovenian |
|---|---|
| Year of publishing: | 2021 |
| Typology: | 2.09 - Master's Thesis |
| Organization: | UL FMF - Faculty of Mathematics and Physics |
| Publisher: | [A. Šadl Praprotnik] |
| UDC: | 519.6 |
| COBISS: |
76510723
|
| Views: | 735 |
| Downloads: | 97 |
| Average score: | 0 (0 votes) |
| Metadata: |
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Other data
| Secondary language: | English |
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| Secondary title: | Conics and quadics in rational Bézier form |
| Secondary abstract: | The master thesis presents conics as rational Bézier curves of degree 2. Some algebraic and geometric properties of conics, presented in such a form, are discussed. The process of changing the form of a conic, from rational Bézier form to implicit form and vice versa, is described and illustrated. We determine the type of a conic, given in a rational Bézier form, based on its number of singularities. Some special attention goes to circular arcs and circles. We show that the degree of the Bézier curve forming a full circle must be at least 5. Quadrics, presented as rational Bézier surfaces, are also discussed. Special attention goes to the sphere. With the help of the stereographic projection we present the octant of a sphere as a rational triangle Bézier patch and a sixth of a sphere as a rational tensor product Bézier surface. |
| Secondary keywords: | Bézier curves;conics;baricentric coordinates;Bézier surfaces;quadrics;stereographic projection; |
| Type (COBISS): | Master's thesis/paper |
| Study programme: | 0 |
| Thesis comment: | Univ. v Ljubljani, Fak. za matematiko in fiziko, Oddelek za matematiko, Računalništvo in matematika - 2. stopnja |
| Pages: | IX, 53 str. |
| ID: | 13388549 |
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