Laguerre Geometry Research Articles

Spinor Representation in Isotropic 3-Space via Laguerre Geometry

We give a detailed description of the geometry of isotropic space, in parallel to those of Euclidean space within the realm of Laguerre geometry. After developing basic surface theory in isotropic space, we define spin transformations, directly leading to the spinor representation of conformal surfaces in isotropic space. As an application, we obtain the Weierstrass-type representation for zero mean curvature surfaces, and the Kenmotsu-type representation for constant mean curvature surfaces, allowing us to construct many explicit examples.

Reconstruction of Single and Dual Phase Microstructures of EN3 and EN8 Steels Using Voronoi Diagram in Laguerre Geometry

Reconstruction of Single and Dual Phase Microstructures of EN3 and EN8 Steels Using Voronoi Diagram in Laguerre Geometry

Discrete Weierstrass-Type Representations

Discrete Weierstrass-type representations yield a construction method in discrete differential geometry for certain classes of discrete surfaces. We show that the known discrete Weierstrass-type representations of certain surface classes can be viewed as applications of the \(\varOmega \)-dual transform to lightlike Gauss maps in Laguerre geometry. From this construction, further Weierstrass-type representations arise. As an application of the techniques we develop, we show that all discrete linear Weingarten surfaces of Bryant or Bianchi type locally arise via Weierstrass-type representations from discrete holomorphic maps.

Modeling of Polycrystalline Material Microstructure with 3D Grain Boundary Based on Laguerre-Voronoi Tessellation.

Voronoi tessellations are shown to be statistically representative of polycrystalline microstructures, which have been widely accepted for the modeling of microstructures of metallurgic and ceramic materials. In this paper, a new implementation of the Voronoi diagram in Laguerre geometry is presented for the generation of numerical models of polycrystalline microstructures, where the size and shape of the grains can be controlled, and the 3D grain boundaries can be modeled with a specified thickness. The distribution of grain sizes in the models is fitted to a lognormal distribution, compared with the normal distribution in the Voronoi tessellation methods. Finally, statistical analyses of grain face and grain size distribution are performed with the models, and the macroscopic elastic properties of polycrystalline ceramic materials are simulated to verify the capability of the presented method.

Generation of stochastic cellular structures with anisotropic cell characteristics on the basis of ellipsoid packings

In the present study, an algorithm to generate anisotropic polygonal cellular structures based on ellipsoid packing is presented. The aim of this process is to obtain structures with different-sized cells which have significantly non-spherical aspect ratios and different orientations in space. The developed method is an extension of the classical Voronoï process in Laguerre geometry which is well-suited to define cell structures with an optimum surface to volume ratio. Different examples show that with this new space division process a number of cellular structures with extremely different characteristics of their cell shape, size and orientation in space can be computed. The generation of such different cellular structures is a useful tool in the multiscale analysis of anisotropic microstructured media with uncertain microstructures exhibiting microstructural anisotropies such as process-related anisotropic structural foams or anisotropic metallic grain agglomerates.

On congruences of Laguerre cycles and their envelopes

In the cyclographic model of Laguerre geometry, the isotropy projection establishes a correspondence between submanifolds in the Lorentz (n + 1)-space and oriented hypersphere congruences in the Euclidean n-space. In this paper, we investigate the relation between the Lorentzian geometry of the sphere congruences and the Euclidean geometry of the corresponding oriented envelopes. We introduce the notion and discuss the existence of marginally outer trapped submanifolds associated to an (n − 1)-dimensional Legendre submanifold of the unit tangent bundle over the Euclidean n-space. This extends the notion of Laguerre–Gauss map of an oriented surface. Two special classes of marginally outer trapped surfaces are characterized in terms of their oriented envelopes. The Lorentzian geometry of several families of spheres whose envelopes are well-known surfaces in the Euclidean three-space is investigated.

Characterizing envelopes of moving rotational cones and applications in CNC machining

Motivated by applications in CNC machining, we provide a characterization of surfaces which are enveloped by a one-parametric family of congruent rotational cones. As limit cases, we also address ruled surfaces and their offsets. The characterizations are higher order nonlinear PDEs generalizing the ones by Gauss and Monge for developable surfaces and ruled surfaces, respectively. The derivation includes results on local approximations of a surface by cones of revolution, which are expressed by contact order in the space of planes. To this purpose, the isotropic model of Laguerre geometry is used as there rotational cones correspond to curves (isotropic circles) and higher order contact is computed with respect to the image of the input surface in the isotropic model. Therefore, one studies curve-surface contact that is conceptually simpler than the surface-surface case. We show that, in a generic case, there exist at most six positions of a fixed rotational cone that have third order contact with the input surface. These results are themselves of interest in geometric computing, for example in cutter selection and positioning for flank CNC machining.

Principal symmetric meshes

The isolines of principal symmetric surface parametrizations run symmetrically to the principal directions. We describe two discrete versions of these special nets/quad meshes which are dual to each other and show their usefulness for various applications in the context of fabrication and architectural design. Our discretization of a principal symmetric mesh comes naturally with a family of spheres, the so-called Meusnier and Mannheim spheres. In our representation of principal symmetric meshes, we have direct control over the radii of theses spheres and the intersection angles of the parameter lines. This facilitates tasks such as generating Weingarten surfaces including constant mean curvature surfaces and minimal surfaces. We illustrate the potential of Weingarten surfaces for paneling doubly curved freeform facades by significantly reducing the number of necessary molds. Moreover, we have direct access to curvature adaptive tool paths for cylindrical CNC milling with circular edges as well as flank milling with rotational cones. Furthermore, the construction of curved support structures from congruent circular strips is easily managed by constant sphere radii. The underlying families of spheres are in a natural way discrete curvature spheres in analogy to smooth Möbius and Laguerre geometry which further leads to a novel discrete curvature theory for principal symmetric meshes.

A new solution of Apollonius’ problem based on stereographic projections of Möbius and Laguerre planes

In this paper we give a new proof of Apollonius’ problem based on the stereographic projection in spherical model of Mobius geometry and cylinder model of Laguerre geometry.

Surfaces in Laguerre geometry

This exposition gives an introduction to the theory of surfaces in Laguerre geometry and surveys some significant results concerning three important classes of surfaces in Laguerre geometry, namely $L$-isothermic, $L$-minimal, and generalized $L$-minim

New insights on marginally trapped surfaces: The hedgehog theory point of view

In this paper, we will try to argue and to show through fundamental examples that (a very huge class of) marginally trapped surfaces arise naturally from a ‘lightlike co-contact structure’, exactly in the same way as Legendrian fronts arise from a contact one (by projection of a Legendrian submanifold to the base of a Legendrian fibration), and that there is an adjunction relationship between both notions. We especially focus our interest on marginally trapped hedgehogs and study their relationships with Laguerre geometry and Brunn–Minkowski theory.

Polynomial conserved quantities of Lie applicable surfaces

Using the gauge theoretic approach for Lie applicable surfaces, we characterise certain subclasses of surfaces in terms of polynomial conserved quantities. These include isothermic and Guichard surfaces of conformal geometry and L-isothermic surfaces of Laguerre geometry. In this setting one can see that the well known transformations available for these surfaces are induced by the transformations of the underlying Lie applicable surfaces. We also consider linear Weingarten surfaces in this setting and develop a new Bäcklund-type transformation for these surfaces.

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.

Unifying Theory of Pythagorean-Normal Surfaces Based on Geometric Algebra

Pythagorean-normal (PN) surfaces, defined as rational surfaces admitting rational offsets, are important for industrial Computer-aided Design. Traditionally PN surfaces are considered from the point of view of Laguerre geometry, using three main models: cyclographic model, Blaschke cylinder or isotropic model. We propose a unifying formalism to deal with PN surfaces: all these models are embedded into one ambient pseudo-Euclidean space \({{\mathbb R}^{4, 2}}\), that is known as a model for Lie sphere geometry. Various relations between different models are described in terms of closed formulas in the geometric algebra \({{\mathcal Cl}(4, 2)}\) and illustrated by examples of applications.

Circle Complexes and the Discrete CKP Equation

In the spirit of Klein's Erlangen Program, we investigate the geometric and algebraic structure of fundamental line complexes and the underlying privileged discrete integrable system for the minors of a matrix which constitute associated Plucker coordinates. Particular emphasis is put on the restriction to Lie circle geometry which is intimately related to the master dCKP equation of discrete integrable systems theory. The geometric interpretation, construction and integrability of fundamental line complexes in Mobius, Laguerre and hyperbolic geometry are discussed in detail. In the process, we encounter various avatars of classical and novel incidence theorems and associated cross- and multi-ratio identities for particular hypercomplex numbers. This leads to a discrete integrable equation which, in the context of Mobius geometry, governs novel doubly hexagonal circle patterns.

Evolutes of plane curves and null curves in Minkowski 3-space

We use the isotropic projection of Laguerre geometry in order to establish a correspondence between plane curves and null curves in the Minkowski 3-space. We describe the geometry of null curves (Cartan frame, pseudo-arc parameter, pseudo-torsion, pairs of associated curves) in terms of the curvature of the corresponding plane curves. This leads to an alternative description of all plane curves which are Laguerre congruent to a given one.

Tessellation Methods for Modeling the Material Structure

Tessellation methods are a relatively new approach for modeling the structure of a material. In this paper, such structures are interpreted as sphere packing models, where molecules and atoms represent spheres of equal or different size. Based on the review of the literature, it is shown that the tessellation approach is a powerful method for modeling and simulating such structures with desirable metric and topological properties. Two basic tessellation methods are considered more in detail: the Delaunay tessellation and the Voronoi diagram in Laguerre geometry, as well as some of their generalizations. The principal concepts of both tessellation methods are briefly explained for a better understanding of the application details. It is noted that packing models created by tessellation methods are not based on the use of the gravity camp effect, which is a difference to numerical and mathematic programming modeling approaches. Therefore, tessellation methods permit the development of structures without taking into account the gravitation, what is important for modeling the structure on the microscopic and nano levels, where the influence of the gravitation is studied insufficiently. A review of the related literature is given, focusing on the details of the tessellation method and the particle size distribution.

Divisible designs, Laguerre geometry, and beyond

In this paper, we bring together design theory and projective geometry over a ring. Both disciplines are well established, but the results on the interaction between them are rare and are scattered over the literature. Our main goal is to present the basics from either side, to develop, or at least sketch, the principal connections between them, and to make recommendations for further reading.

The Second Variational Formula for Laguerre Minimal Hypersurfaces in $${\mathbb {R}^n}$$

Li and Wang (Manuscr Math 122(1):73–95, 2007) presented Laguerre geometry for hypersurfaces in \({\mathbb{R}^{n}}\) and calculated the first variational formula of the Laguerre functional by using Laguerre invariants. In this paper we present the second variational formula for Laguerre minimal hypersurfaces. As an application of this variational formula we give the standard examples of Laguerre minimal hypersurfaces in \({\mathbb{R}^{n}}\) and show that they are stable Laguerre minimal hypersurfaces. Using this second variational formula we can prove that a surface with vanishing mean curvature in \({\mathbb{R}^{3}_{0}}\) is Laguerre equivalent to a stable Laguerre minimal surface in \({\mathbb{R}^{3}}\) under the Laguerre embedding. This example of stable Laguerre minimal surface in \({\mathbb{R}^{3}}\) is different from the one Palmer gave in (Rend Mat Appl 19(2):281–293, 1999).

Assessment of material uncertainties in solid foams based on local homogenization procedures

The present study is concerned with a numerical prediction and assessment of uncertainties in the macroscopic material properties of solid foams. The material properties are determined by means of a homogenization analysis considering a large scale representative volume element. The microstructure for the representative volume element is determined randomly using a Voronoï tesselation in Laguerre geometry with prescribed cell size distribution. For assessment of the scatter in the effective material response, the homogenization scheme is applied to subsets of the large scale representative volume element. By this means, an interrelation between the local microstructural characteristics and the local mesoscopic material response is established. In a first approach, the individual cells of the foam microstructure are employed as testing volume elements. As an alternative, a moving window technique is applied. The sets of homogenization results obtained by both approaches are evaluated by stochastic methods. For the local effective properties, a distinct scatter is observed. The results in both cases reveal that the local porosity is the most important influence parameter. For the microstructures investigated, only weak local correlations of the effective stiffnesses with a rapid spatial decay of the correlation is observed.