Krivulje s pitagorejskim hodografom in interpolacija : magistrsko delo

Kramer, Sabina (author), Žagar, Emil (mentor)

Na začetku bomo definirali osnovne lastnosti ravninskih parametričnih krivulj, kot so tangenta, ukrivuljenost, normala in paralelna krivulja. Potem se bomo posvetili polinomom v Bernsteinovi bazi. Na podlagi teh polinomov bomo definirali Bézierjeve krivulje in predstavili pomen kontrolnega poligona. Sledila bo interpolacija s kubičnimi Bézierjevimi krivuljami. Definirali bomo krivulje s pitagorejskim hodografom (PH krivulje). Opisali bomo njihove glavne lastnosti in predstavili formule za izračun kontrolnih točk. Nato se bomo ukvarjali z interpolacijo s PH krivuljami stopnje 5. Predstavili bomo kriterij za izbiro najboljše rešitve in konstruirali zlepke, ki jih bomo primerjali s kubičnimi zlepki. Konstruirali bomo paralelne krivulje in predstavili metodo za obrezovanje teh krivulj.

• Language: Slovenian
• Keywords: parametrična krivulja, tangentni vektor, ukrivljenost, paralelna krivulja, Bernsteinova baza, Bézierjeva krivulja, pitagorejski hodograf, PH krivulja, Hermiteova interpolacija, zlepek
• Work type: Master's thesis/paper
• Typology: 2.09 - Master's Thesis
• Organization: FMF - Faculty of Mathematics and Physics
• Year: 2018
• PID: 20.500.12556/RUL-103997
• UDC: 519.6
• COBISS.SI-ID: 18459225
• Publication date in RUL: 30.09.2018

KRAMER, Sabina, 2018, Krivulje s pitagorejskim hodografom in interpolacija : magistrsko delo [online]. Master’s thesis. [Accessed 10 October 2025]. Retrieved from: https://repozitorij.uni-lj.si/IzpisGradiva.php?lang=eng&id=103997

Secondary language

• Language: English
• Title: Pythagorean-Hodograph curves and interpolation

Abstract

We will define basic parametric planar curve properties, like tangent vector, curvature, normal vector and offset. Then, we will describe polynomials in the Bernstein basis and use that concept for defining Bézier curves and control polygon. Interpolation with cubic Bézier will follow. We will define pythagorean-hodograph (PH) curves, describe their main properties and calculate control points. We will interpolate given data with the PH quintics and show a criteria for choosing the best solution. We will construct PH quintic splines and compare them to the ordinary cubic splines. We will finish with constructing offset curves and describe trimming procedure.

• Keywords: parametric curve, tangent vector, curvature, offset curve, Bernstein basis, Bézier curve, pythagorean hodograph, PH curve, Hermite interpolation, spline