Computation of Yvon-Villarceau circles on Dupin cyclides and construction of circular edge right triangles on tori and Dupin cyclides

Lionel Garnier, Hichem Barki, Sebti Foufou, Loic Puech

    Research output: Contribution to journalArticle

    Abstract

    Ring Dupin cyclides are non-spherical algebraic surfaces of degree four that can be defined as the image by inversion of a ring torus. They are interesting in geometric modeling because: (1) they have several families of circles embedded on them: parallel, meridian, and Yvon-Villarceau circles, and (2) they are characterized by one parametric equation and two equivalent implicit ones, allowing for better flexibility and easiness of use by adopting one representation or the other, according to the best suitability for a particular application. These facts motivate the construction of circular edge triangles lying on Dupin cyclides and exhibiting the aforementioned properties. Our first contribution consists in an analytic method for the computation of Yvon-Villarceau circles on a given ring Dupin cyclide, by computing an adequate Dupin cyclide-torus inversion and applying it to the torus-based equations of Yvon-Villarceau circles. Our second contribution is an algorithm which, starting from three arbitrary 3D points, constructs a triangle on a ring torus such that each of its edges belongs to one of the three families of circles on a ring torus: meridian, parallel, and Yvon-Villarceau circles. Since the same task of constructing right triangles is far from being easy to accomplish when directly dealing with cyclides, our third contribution is an indirect algorithm which proceeds in two steps and relies on the previous one. As the image of a circle by a carefully chosen inversion is a circle, and by constructing different images of a right triangle on a ring torus, the indirect algorithm constructs a one-parameter family of 3D circular edge triangles lying on Dupin cyclides.

    Original languageEnglish (US)
    Pages (from-to)1689-1709
    Number of pages21
    JournalComputers and Mathematics with Applications
    Volume68
    Issue number12
    DOIs
    StatePublished - Jan 1 2014

    Fingerprint

    Right-angled triangle
    Torus
    Circle
    Ring
    Dupin cyclide
    Meridian
    Triangle
    Inversion
    Parametric equations
    Geometric Modeling
    Algebraic Surfaces
    Flexibility
    Computing

    Keywords

    • Circular edge right triangle
    • Inversion
    • Ring Dupin cyclide
    • Ring torus
    • Yvon-Villarceau circle

    ASJC Scopus subject areas

    • Modeling and Simulation
    • Computational Theory and Mathematics
    • Computational Mathematics

    Cite this

    Computation of Yvon-Villarceau circles on Dupin cyclides and construction of circular edge right triangles on tori and Dupin cyclides. / Garnier, Lionel; Barki, Hichem; Foufou, Sebti; Puech, Loic.

    In: Computers and Mathematics with Applications, Vol. 68, No. 12, 01.01.2014, p. 1689-1709.

    Research output: Contribution to journalArticle

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