1,283 research outputs found

    An application of Groebner bases to planarity of intersection of surfaces

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    In this paper we use Groebner bases theory in order to determine planarity of intersections of two algebraic surfaces in R3{\bf R}^3. We specially considered plane sections of certain type of conoid which has a cubic egg curve as one of the directrices. The paper investigates a possibility of conic plane sections of this type of conoid

    The symmetric, D-invariant and Egorov reductions of the quadrilateral lattice

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    We present a detailed study of the geometric and algebraic properties of the multidimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar; the discrete analogue of a conjugate net) and of its basic reductions. To make this study, we introduce the notions of forward and backward data, which allow us to give a geometric meaning to the tau-function of the lattice, defined as the potential connecting these data. Together with the known circular lattice (a lattice whose elementary quadrilaterals can be inscribed in circles; the discrete analogue of an orthogonal conjugate net) we introduce and study two other basic reductions of the quadrilateral lattice: the symmetric lattice, for which the forward and backward data coincide, and the D-invariant lattice, characterized by the invariance of a certain natural frame along the main diagonal. We finally discuss the Egorov lattice, which is, at the same time, symmetric, circular and D-invariant. The integrability properties of all these lattices are established using geometric, algebraic and analytic means; in particular we present a D-bar formalism to construct large classes of such lattices. We also discuss quadrilateral hyperplane lattices and the interplay between quadrilateral point and hyperplane lattices in all the above reductions.Comment: 48 pages, 6 figures; 1 section added, to appear in J. Geom. & Phy

    On organizing principles of Discrete Differential Geometry. Geometry of spheres

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    Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem discussed in this survey is a discretization of curvature line parametrized surfaces in Lie geometry. We find a discretization of curvature line parametrization which unifies the circular and conical nets by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is slightly changed and umbilic points are discusse

    Miscellaneous properties of embeddings of line, total and middle graphs

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    Chartrand et al. (J. Combin. Theory Ser. B 10 (1971) 12–41) proved that the line graph of a graph G is outerplanar if and only if the total graph of G is planar. In this paper, we prove that these two conditions are equivalent to the middle graph of G been generalized outerplanar. Also, we show that a total graph is generalized outerplanar if and only if it is outerplanar. Later on, we characterize the graphs G such that Full-size image (<1 K) is planar, where Full-size image (<1 K) is a composition of the operations line, middle and total graphs. Also, we give an algorithm which decides whether or not Full-size image (<1 K) is planar in an Full-size image (<1 K) time, where n is the number of vertices of G. Finally, we give two characterizations of graphs so that their total and middle graphs admit an embedding in the projective plane. The first characterization shows the properties that a graph must verify in order to have a projective total and middle graph. The second one is in terms of forbidden subgraphs
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