58,206 research outputs found
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
Discrete asymptotic nets and W-congruences in Plucker line geometry
The asymptotic lattices and their transformations are studied within the line
geometry approach. It is shown that the discrete asymptotic nets are
represented by isotropic congruences in the Plucker quadric. On the basis of
the Lelieuvre-type representation of asymptotic lattices and of the discrete
analog of the Moutard transformation, it is constructed the discrete analog of
the W-congruences, which provide the Darboux-Backlund type transformation of
asymptotic lattices.The permutability theorems for the discrete Moutard
transformation and for the corresponding transformation of asymptotic lattices
are established as well. Moreover, it is proven that the discrete W-congruences
are represented by quadrilateral lattices in the quadric of Plucker. These
results generalize to a discrete level the classical line-geometric approach to
asymptotic nets and W-congruences, and incorporate the theory of asymptotic
lattices into more general theory of quadrilateral lattices and their
reductions.Comment: 28 pages, 4 figures; expanded Introduction, new Section, added
reference
Ruled Laguerre minimal surfaces
A Laguerre minimal surface is an immersed surface in the Euclidean space
being an extremal of the functional \int (H^2/K - 1) dA. In the present paper,
we prove that the only ruled Laguerre minimal surfaces are up to isometry the
surfaces R(u,v) = (Au, Bu, Cu + D cos 2u) + v (sin u, cos u, 0), where A, B, C,
D are fixed real numbers. To achieve invariance under Laguerre transformations,
we also derive all Laguerre minimal surfaces that are enveloped by a family of
cones. The methodology is based on the isotropic model of Laguerre geometry. In
this model a Laguerre minimal surface enveloped by a family of cones
corresponds to a graph of a biharmonic function carrying a family of isotropic
circles. We classify such functions by showing that the top view of the family
of circles is a pencil.Comment: 28 pages, 9 figures. Minor correction: missed assumption (*) added to
Propositions 1-2 and Theorem 2, missed case (nested circles having nonempty
envelope) added in the proof of Pencil Theorem 4, missed proof that the arcs
cut off by the envelope are disjoint added in the proof of Lemma
Discrete constant mean curvature nets in space forms: Steiner's formula and Christoffel duality
We show that the discrete principal nets in quadrics of constant curvature
that have constant mixed area mean curvature can be characterized by the
existence of a K\"onigs dual in a concentric quadric.Comment: 12 pages, 10 figures, pdfLaTeX (plain pdfTeX source included as bak
file
The Ribaucour transformation in Lie sphere geometry
We discuss the Ribaucour transformation of Legendre maps in Lie sphere
geometry. In this context, we give a simple conceptual proof of Bianchi's
original Permutability Theorem and its generalisation by Dajczer--Tojeiro. We
go on to formulate and prove a higher dimensional version of the Permutability
Theorem. It is shown how these theorems descend to the corresponding results
for submanifolds in space forms.Comment: v2: Introduction expanded and references added. 20 pages, 4
Postscript figure
Orthogonal nets and Clifford algebras
A Clifford algebra model for M"obius geometry is presented. The notion of
Ribaucour pairs of orthogonal systems in arbitrary dimensions is introduced,
and the structure equations for adapted frames are derived. These equations are
discretized and the geometry of the occuring discrete nets and sphere
congruences is discussed in a conformal setting. This way, the notions of
``discrete Ribaucour congruences'' and ``discrete Ribaucour pairs of orthogonal
systems'' are obtained --- the latter as a generalization of discrete
orthogonal systems in Euclidean space. The relation of a Cauchy problem for
discrete orthogonal nets and a permutability theorem for the Ribaucour
transformation of smooth orthogonal systems is discussed.Comment: Plain TeX, 16 pages, 4 picture
Atom-Based Geometrical Fingerprinting of Conformal Two-Dimensional Materials
The shape of two-dimensional materials plays a significant role on their chemical and physical properties. Two-dimensional materials are basic meshes that are formed by mesh points (vertices) given by atomic positions, and connecting lines (edges) between points given by chemical bonds. Therefore the study of local shape and geometry of two-dimensional materials is a fundamental prerequisite to investigate physical and chemical properties. Hereby the use of discrete geometry to discuss the shape of two-dimensional materials is initiated.
The local geometry of a surface embodied in 3D space is determined using four invariant numbers from the metric and curvature tensors which indicates how much the surface is stretched and curved under a deformation as compared to a reference pre-deformed conformation.
Many different disciplines advance theories on conformal two-dimensional materials by relying on continuum mechanics and fitting continuum surfaces to the shape of conformal two-dimensional materials. However two-dimensional materials are inherently discrete. The continuum models are only applicable when the size of two-dimensional materials is significantly large and the deformation is less than a few percent. In this research, the knowledge of discrete differential geometry was used to tell the local shape of conformal two-dimensional materials. Three kind of two-dimensional materials are discussed: 1) one atom thickness structures such as graphene and hexagonal boron nitride; 2) high and low buckled 2D meshes like stanene, leadene, aluminum phosphate; and, 3) multi layer 2D materials such as Bi2Se3 and WSe2. The lattice structures of these materials were created by designing a mechanical model - the mechanical model was devised in the form of a Gaussian bump and density-functional theory was used to inform the local height; and, the local geometries are also discussed
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