q-Bernsteinovi polinomi in q-Bézierjeve krivulje
delo diplomskega seminarja
Abstract
V teoriji aproksimacije in v računalniško podprtem geometrijskem oblikovanju je pomemben problem poiskati aproksimacije točk v prostoru s pomočjo polinomov in parametričnih polinomskih krivulj. Možno rešitev nam ponujajo Bernsteinovi aproksimacijski polinomi, ki so bili prvič predstavljeni pri dokazu Stone-Weirestrassovega izreka, ter Bézierjeve krivulje, ki so osnovni objekti pri modeliranju s krivuljami. V delu bo opisana njihova posplošitev s pomočjo parametra $q$.
Predstavljeni bodo Bernsteinovi bazni polinomi in njihova posplošitev na $q$-Bernsteinove bazne polinome, s pomočjo katerih definiramo $q$-Bernsteinove aproksimacijske polinome ter $q$-Bézierjeve krivulje. Izpeljane bodo osnovne lastnosti tako za standardni primer, ko je $q$=1, kot tudi za splošen primer.
Med drugim bo predstavljena posplošitev de Casteljaujevega algoritma, ki je numerično stabilen algoritem za računanje točk na Bézierjevi krivulji, posplošitev postopka višanja stopnje krivulje ter računanje odvodov q-Bézierjevih krivulj.
Teoretični rezultati bodo ilustrirani z različnimi numeričnimi primeri.
Keywords
q-Bernsteinov polinom;q-Bézierjeva krivulja;totalno pozitivne baze;de Casteljaujev algoritem;
Data
| Language: | Slovenian |
|---|---|
| Year of publishing: | 2020 |
| Typology: | 2.11 - Undergraduate Thesis |
| Organization: | UL FMF - Faculty of Mathematics and Physics |
| Publisher: | [M. Kokošinek] |
| UDC: | 519.6 |
| COBISS: |
58244611
|
| Views: | 933 |
| Downloads: | 127 |
| Average score: | 0 (0 votes) |
| Metadata: |
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Other data
| Secondary language: | English |
|---|---|
| Secondary title: | q-Bernstein polynomials and q-Bézier curves |
| Secondary abstract: | In approximation theory and computer aided geometric design an important problem is to find the approximation of points with the use of polynomials and parametric polynomial curves. This can be achieved with Bernstein aproximation polynomials, which were introduced in the proof of Stone-Weierstrass theorem, and with Bézier curves, which are the basic objects in curve modelling. The work focuses on their generalization which can be achieved with the introduction of parameter $q$. Bernstein basis polynomials and their generalization to $q$-Bernstein basis polynomials, with which we can define $q$-Bernstein approximation polynomials and $q$-Bézier curves, are introduced. The elementary characteristics for the standard example, when $q=1$, are derived, as well as the elementary characteristics of a more general example. The generalization of de Casteljau algorithm, which is a stable algorithm for the calculation of points on the Bézier curve is also presented, as well as the generalisation of the curve degree elevation procedure and calculation of derivatives of the $q$-Bézier curves. Theoretic examples are illustrated with a variety of numerical examples. |
| Secondary keywords: | q-Bernstein polynomial;q-Bézier curve;totally positive bases;de Casteljau algorithm; |
| 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, Finančna matematika - 1. stopnja |
| Pages: | 28 str. |
| ID: | 12114459 |
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