1,283 research outputs found
An application of Groebner bases to planarity of intersection of surfaces
In this paper we use Groebner bases theory in order to determine planarity of
intersections of two algebraic surfaces in . 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
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
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
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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