Interpolacija nad trikotniki : delo diplomskega seminarja

Holc, Lea (author), Knez, Marjetka (mentor)

V delu diplomskega seminarja se ukvarjamo z interpolacijo funkcij dveh spremenljivk nad trikotno domeno. Za zapis interpolacijskih točk v trikotniku uporabimo baricentrične koordinate namesto kartezičnih. Definiramo Bernsteinove bazne polinome, za katere dokažemo, da so baza prostora polinomov dveh spremenjivk skupne stopnje največ $n$ in spoznamo nekaj njihovih ključnih lastnosti. Vpeljemo Bernstein-Bézierjevo obliko zapisa polinoma, pokažemo, da je taka oblika stabilna v neskončni normi in spoznamo de Casteljaujev algoritem za računanje vrednosti polinoma v tej obliki. V nadaljevanju navedemo formule za izračun Bézierjevih ordinat za interpolacijo zveznih funkcij s Hermitovimi polinomi stopnje $n = 3$, ogledamo pa si tudi interpolacijo s konstrukcijo Argyrisovega elementa. Za lažje razumevanje oba primera utemeljimo s praktičnima primeroma. Obravnavamo tudi interpolacijo nad dvema sosednjima trikotnikoma ter zapišemo in dokažemo pogoje, ki nam zagotovijo $C^1$ gladkost med sosednjima trikotnima ploskvama.

• Language: Slovenian
• Keywords: interpolacija, baricentrične koordinate, Bernsteinovi bazni polinomi, Argyrisov element, Bézierjeve ordinate
• Work type: Final seminar paper
• Typology: 2.11 - Undergraduate Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2024
• PID: 20.500.12556/RUL-158657
• UDC: 519.6
• COBISS.SI-ID: 199359235
• Publication date in RUL: 19.06.2024

HOLC, Lea, 2024, Interpolacija nad trikotniki : delo diplomskega seminarja [online]. Bachelor’s thesis. [Accessed 10 October 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=158657

Secondary language

• Language: English
• Title: Interpolation over triangular domain

Abstract

In this thesis we consider the interpolation of bivariate functions defined over a triangular domain. Barycentric coordinates are used instead of Cartesian coordinates to compute the interpolation points in the triangle. Additionally, Bernstein basis polynomials with some of their key properties are defined and proven to be the basis of the space of polynomials of two variables of total degree at most $n$. We also introduce the Bernstein-Bézier form for polynomials, show that such a form is stable in the infinite norm and define de Casteljau's algorithm for computing values of a polynomial in such a form. Furthermore, formulas for computing Bézier ordinates for the interpolation of continuous functions with Hermite polynomials of degree $n = 3$ and $n=5$, which we call Argyris elements, are derived. Both cases are illustrated with practical examples to make it easier to understand. We also consider interpolation over two adjacent triangles, and we derive and prove the conditions which ensure $C^1$ smoothness between two adjacent triangular patches.

• Keywords: interpolation, barycentric coordinates, Bernstein basis polynomials, Argyris element, Bézier ordinates