Elementary Quadrilaterals Research Articles

Nets of Lines with the Combinatorics of the Square Grid and with Touching Inscribed Conics

In the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially neighbouring quadrilaterals have the same touching point on their common edge-line. We suggest that these nets are a natural projective generalisation of incircular nets. It is shown that these nets are planar Koenigs nets. Moreover, we show that general Koenigs nets are characterised by the existence of a 1-parameter family of touching inscribed conics. It is shown that the lines of any grid of quadrilaterals with touching inscribed conics are tangent to a common conic. These grids can be constructed via polygonal chains that are inscribed in conics. The special case of billiards in conics corresponds to incircular nets.

Multi-Nets. Classification of Discrete and Smooth Surfaces with Characteristic Properties on Arbitrary Parameter Rectangles

We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on arbitrary parameter rectangles. For discrete planar quadrilateral nets, circular nets, $$Q^*$$-nets and conical nets we obtain a characterization of the corresponding discrete multi-nets. In the limit these discrete nets lead to some classical classes of smooth surfaces. Furthermore, we propose to use the characterized discrete nets as discrete extensions for the nets to obtain structure preserving subdivision schemes.

Incircular nets and confocal conics

We consider congruences of straight lines in a plane with the combinatorics of the square grid, with all elementary quadrilaterals possessing an incircle. It is shown that all the vertices of such nets (we call them incircular or IC-nets) lie on confocal conics. Our main new results are on checkerboard IC-nets in the plane. These are congruences of straight lines in the plane with the combinatorics of the square grid, combinatorially colored as a checkerboard, such that all black coordinate quadrilaterals possess inscribed circles. We show how this larger class of IC-nets appears quite naturally in Laguerre geometry of oriented planes and spheres and leads to new remarkable incidence theorems. Most of our results are valid in hyperbolic and spherical geometries as well. We present also generalizations in spaces of higher dimension, called checkerboard IS-nets. The construction of these nets is based on a new 9 9 inspheres incidence theorem.

Discretization of asymptotic line parametrizations using hyperboloid surface patches

Two-dimensional affine A-nets in 3-space are quadrilateral meshes that discretize surfaces parametrized along asymptotic lines. The characterizing property of A-nets is planarity of vertex stars, so for generic A-nets the elementary quadrilaterals are skew. We classify the simply connected affine A-nets that can be extended to continuously differentiable surfaces by gluing hyperboloid surface patches into the skew quadrilaterals. The resulting surfaces are called "hyperbolic nets" and are a novel piecewise smooth discretization of surfaces parametrized along asymptotic lines. It turns out that a simply connected affine A-net has to satisfy one combinatorial and one geometric condition to be extendable - all vertices have to be of even degree and all quadrilateral strips have to be "equi-twisted". Furthermore, if an A-net can be extended to a hyperbolic net, then there exists a 1-parameter family of such C^1-surfaces. It is briefly explained how the generation of hyperbolic nets can be implemented on a computer. The article uses the projective model of Pluecker geometry to describe A-nets and hyperboloids.

Integrable Discrete Nets in Grassmannians

We consider discrete nets in Grassmannians $\mathbb{G}^d_r$ which generalize Q-nets (maps $\mathbb{Z}^N\to\mathbb{P}^d$ with planar elementary quadrilaterals) and Darboux nets ($\mathbb{P}^d$-valued maps defined on the edges of $\mathbb{Z}^N$ such that quadruples of points corresponding to elementary squares are all collinear). We give a geometric proof of integrability (multidimensional consistency) of these novel nets, and show that they are analytically described by the noncommutative discrete Darboux system.

Discrete Koenigs Nets and Discrete Isothermic Surfaces

We discuss discretization of Koenigs nets (conjugate nets with equal Laplace invariants) and of isothermic surfaces. Our discretization is based on the notion of dual quadrilaterals: two planar quadrilaterals are called dual if their corresponding sides are parallel, and their noncorresponding diagonals are parallel. Discrete Koenigs nets are defined as nets with planar quadrilaterals admitting dual nets. Several novel geometric properties of discrete Koenigs nets are found; in particular, two-dimensional discrete Koenigs nets can be characterized by coplanarity of the intersection points of diagonals of elementary quadrilaterals adjacent to any vertex; this characterization is invariant with respect to projective transformations. Discrete isothermic nets are defined as circular Koenigs nets. This is a new geometric characterization of discrete isothermic surfaces introduced previously as circular nets with factorized cross-ratios.

Discretization of multidimensional submanifolds associated with spin-valued spectral problems

We present a large family of Spin(p, q)-valued discrete spectral problems. The associated discrete nets generated by the so-called Sym-Tafel formula are circular nets (i.e., all elementary quadrilaterals are inscribed into circles). These nets are discrete analogues of smooth multidimensional immersions in ℝm including isothermic surfaces, Guichard nets, and some other families of orthogonal nets.

THE SELF-ADJOINT 5-POINT AND 7-POINT DIFFERENCE OPERATORS, THE ASSOCIATED DIRICHLET PROBLEMS, DARBOUX TRANSFORMATIONS AND LELIEUVRE FORMULAE

We present some basic properties of two distinguished discretizations of elliptic operators: the self-adjoint 5-point and 7-point schemes on a two dimensional lattice. We first show that they allow us to solve Dirichlet boundary value problems; then we present their Moutard transformations (distinguished examples of transformation of Darboux type in two dimensions). Finally we construct their Lelieuvre formulae and we show that, at the level of the normal vector and in full analogy with their continuous counterparts, the self-adjoint 5-point scheme characterizes a two dimensional quadrilateral lattice (a lattice whose elementary quadrilaterals are planar), while the self-adjoint 7-point scheme characterizes a generic 2D lattice.

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 analog 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 τ-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 analog of an orthogonal conjugate net) we introduce and study two other basic and independent 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 ∂ ̄ 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.

Transformations of quadrilateral lattices

Motivated by the classical studies on transformations of conjugate nets, we develop the general geometric theory of transformations of their discrete analogs: the multidimensional quadrilateral lattices, i.e., lattices x:ZN→RM, N⩽M, whose elementary quadrilaterals are planar. Our investigation is based on the discrete analog of the theory of the rectilinear congruences, which we also present in detail. We study, in particular, the discrete analogs of the Laplace, Combescure, Lévy, radial, and fundamental transformations and their interrelations. The composition of these transformations and their permutability is also investigated from a geometric point of view. The deep connections between “transformations” and “discretizations” is also investigated for quadrilateral lattices. We finally interpret these results within the ∂̄ formalism.

Three–dimensional integrable lattices in Euclidean spaces: conjugacy and orthogonality

It is shown that the discrete Darboux system, descriptive of conjugate lattices in Euclidean spaces, admits constraints on the (adjoint) eigenfunctions which may be interpreted as discrete orthogonality conditions on the lattices. Thus, it turns out that the elementary quadrilaterals of orthogonal lattices are cyclic. Orthogonal lattices on lines, planes and spheres are discussed and the underlying integrable systems in one, two and three dimensions are derived explicitly. A discrete analogue of Bianchi's Ribaucour transformation is set down and particular orthogonal lattices are given. As a by–product, discrete Dini surfaces are obtained.

―∂-Reductions of the Multidimensional Quadrilateral Lattice. The Multidimensional Circular Lattice

We apply a recently introduced [21, 15] reduction method, based on the ―∂ - dressing, to construct a large class of integrable reductions of the equations characterizing the multidimensional quadrilateral lattice (an N-dimensional lattice in ℝM, N≤M, whose elementary quadrilaterals are planar and whose continuous limit describes submanifolds parametrized by conjugate lines [11]). We also show that, generically, in the limit of the small lattice parameter, half of these reductions lead to the Darboux equations for symmetric fields and the second half lead to the generalized Lame equations describing N-dimensional submanifolds of ?M parametrized by conjugate orthogonal systems of coordinates. We finally show that a distinguished example of the second class of reductions corresponds to the multidimensional circular lattice (an N-dimensional lattice in ?M, N≤M, whose elementary quadrilaterals are inscribed in circles [7]).

The integrable discrete analogues of orthogonal coordinate systems are multi-dimensional circular lattices

We show that the integrable discrete analogues of the Lamé orthogonal systems of coordinates are given by multidimensional circular lattices, i.e. by multidimensional lattices whose elementary quadrilaterals are inscribed in circles.