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V magistrskem delu se ukvarjamo s krivuljami s pitagorejskim hodografom, polinomskimi vijačnicami ter DPH-krivuljami. Na začetku se bomo seznanili z osnovnimi lastnostmi parametrično podanih prostorskih krivulj, Bézierjevimi krivuljami, Bernsteinovimi polinomi ter vektorskim prostorom kvaternionov. Nadaljevali bomo z obravnavo krivulj s pitagorejskim hodografom, spoznali nekaj lastnosti teh krivulj in jih izrazili s pomočjo kvaternionov ter Hopfove preslikave. Vpeljali bomo pojem polinomskih DPH-krivulj in klasificirali različne tipe teh krivulj pri nizkih stopnjah. Raziskali bomo povezavo teh krivulj z vijačnimi krivuljami, ki imajo polinomsko parametrizacijo in si ogledali različne postopke, s katerimi lahko konstruiramo različne tipe DPH-krivulj. Ogledali si bomo še pogoje za obstoj nevijačnih DPH-krivulj ter podali nekaj primerov. Za konec bo sledilo še poglavje o Hermitovi interpolaciji z vijačnimi DPH-krivuljami stopnje 5. The topic of the master thesis is a subclass of Pythagorean-hodograph curves, named DPH curves and their relationship with polynomial helical curves. At the beginning, we will get acquainted with the basic properties of parametric spatial curves, Bézier curves, Bernstein polynomials and vector space of quaternions. We will continue with the discussion on Pythagorean-hodograph curves, learn some of the properties of these curves and how to represent them using quaternion and Hopf map form. The concept of polynomial DPH curves will be introduced and classification of different types of these curves at low degrees will be given. We will investigate the similarities between DPH curves and helical curves that have polynomial parametrization and describe various procedures by which we can construct different types of DPH curves. The conditions for the existence of non-helical DPH curves and some examples will be given. Finally, there will be a chapter on the Hermite interpolation by helical DPH curves of degree 5. |
