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Total curvature of space curves and Fáry-Milnor theorem

Nejc Štamcar (2012) Total curvature of space curves and Fáry-Milnor theorem. Diploma thesis.

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    Abstract

    Purpose of diploma thesis is examining several types of curvature of parametrized curves, especially total curvature, as well as presentation and detailed proof of Fáry-Milnor theorem. Thesis is about parametrized curves and their various characteristics. It deals with closed curves, their periodicity, different types of curvature, convexity of curves and knotted curves. It includes both curves in the plane, as well as curves in space R³ and multi-dimensional spaces. In diploma we first discussed about curves in optional n-dimensional spaces. We focused only on the curves, which are given parametrically. We precisely explained, when the curves are regular parametrized and when they are parametrized by arc-length. We continued with research of periodic and closed curves, and with identifying a correlation between these two terms. We have also become familiar with the term of simple closed curves. Then we specifically talked about plane curves and curves in space R³. We examined in detail the convexity, curvature, torsion, orthogonal and Frenet basis of space, the connection between the two types of curvature and the vectors that form Frenet basis. We explained the concept of osculating, normal and rectifying plane. We have investigated total curvature of curves and partially investigated knotted curves. We concluded with the outline of life and work of two important mathematicians (István Fáry, John Milnor), a presentation of theorem that is named by them, as well as with detailed proof of this theorem.

    Item Type: Thesis (Diploma thesis)
    Keywords: ambient isotopy, binormal, bridge number, closed curve, convex curve, curvature, Fáry-Milnor theorem, Frenet basis, Frenet formulas, isotopy, knot, knotted curve, length of a curve, lifting lemma, normal, normal plane, osculating plane, parametrisation, parametrisation by arc-length, period, periodic curve, rectifying plane, regular parametrisation, simple closed curve, star-shaped set, tangent, torsion, total curvature
    Number of Pages: 49
    Language of Content: Slovenian
    Mentor / Comentors:
    Mentor / ComentorsIDFunction
    izr. prof. dr. Matija CenceljMentor
    Link to COBISS: http://www.cobiss.si/scripts/cobiss?command=search&base=50126&select=(ID=9414985)
    Institution: University of Ljubljana
    Department: Faculty of Education, Faculty of mathematics and physics
    Item ID: 1110
    Date Deposited: 02 Oct 2012 11:48
    Last Modified: 02 Oct 2012 11:48
    URI: http://pefprints.pef.uni-lj.si/id/eprint/1110

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