DGD Gallery: Storage, sharing, and publication of digital research data

We describe a project, called the "Discretization in Geometry and Dynamics Gallery", or DGD Gallery for short, whose goal is to store geometric data and to make it publicly available. The DGD Gallery offers an online web service for the storage, sharing, and publication of digital research data.Comment: 19 pages, 8 figures, to appear in "Advances in Discrete Differential Geometry", ed. A. I. Bobenko, Springer, 201

On a discretization of confocal quadrics. I. An integrable systems approach

Confocal quadrics lie at the heart of the system of confocal coordinates (also called elliptic coordinates, after Jacobi). We suggest a discretization which respects two crucial properties of confocal coordinates: separability and all two-dimensional coordinate subnets being isothermic surfaces (that is, allowing a conformal parametrization along curvature lines, or, equivalently, supporting orthogonal Koenigs nets). Our construction is based on an integrable discretization of the Euler-Poisson-Darboux equation and leads to discrete nets with the separability property, with all two-dimensional subnets being Koenigs nets, and with an additional novel discrete analog of the orthogonality property. The coordinate functions of our discrete nets are given explicitly in terms of gamma functions.Comment: 37 pp., 9 figures. V2 is a completely reworked and extended version, with a lot of new materia

On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs

Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics which are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines which are diagonally related to lines of curvature is proven theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory

Diskerete konfokale Quadriken und Schachbrettinkreisnetze

Confocal quadrics constitute a special example of orthogonal coordinate systems. In this cumulative thesis we propose two approaches to the discretization of confocal coordinates, and study the closely related checkerboard incircular nets. First, we propose a discretization based on factorizable solutions to an integrable discretization of the Euler-Poisson-Darboux equation. The constructed solutions are discrete Koenigs nets and feature a novel discrete orthogonality constraint defined on pairs of dual discrete nets, as well as a corresponding discrete isothermicity condition. The coordinate functions of these discrete confocal coordinates are explicitly given in terms of gamma functions. Secondly, we show that classical confocal coordinates and their reparametrizations along coordinate lines are characterized by orthogonality and the factorization property. We use these two properties to propose another discretization of confocal coordinates, while again employing the aforementioned discrete orthogonality constraint. In comparison to the first approach, this definition results in a broader class of nets capturing arbitrary reparametrizations also in the discrete case. We show that these discrete confocal coordinate systems may equivalently be constructed geometrically via polarity with respect to a sequence of classical confocal quadrics. Different sequences correspond to different discrete parametrizations. We give several explicit examples, including parametrizations in terms of Jacobi elliptic functions. A particular example of discrete confocal coordinates in the two-dimensional case is closely related to incircular nets, that is, congruences of straight lines in the plane with the combinatorics of the square grid such that each elementary quadrilateral admits an incircle. Thus, thirdly, we classify and integrate the class of checkerboard incircular nets, which constitute the Laguerre geometric generalization of incircular nets. Further aspects of the novel discrete orthogonality constraint are studied in the introduction of this thesis. These include discrete Lamé coefficients, discrete focal nets, discrete parallel nets, and discrete isothermicity, as well as the relation to pairs of circular and conical nets.Konfokale Quadriken bilden ein Beispiel für orthogonale Koordinatensysteme. In dieser kumulativen Dissertation werden zwei Ansätze zur Diskretisierung konfokaler Koordinaten sowie der Zusammenhang zu Schachbrettinkreisnetzen behandelt. Der erste Ansatz begründet sich auf einer integrablen Diskretisierung der Euler Poisson-Darboux-Gleichung. Die konstruierten Lösungen sind diskrete Koenigs-Netze und durch eine neue diskrete Orthogonalitätsbedingung gekennzeichnet. Die Koordinatenfunktionen sind explizit durch gamma-Funktionen gegeben. Für den zweiten Ansatz zeigen wir zunächst, dass klassische konfokale Koordiatensysteme bis auf Umparametrisierung entlang der Koordinatenlinien durch Orthogonalität und die Faktorsierbarkeit bereits charakterisiert sind. Wir übertragen diese beiden Eigenschaften auf eine weitere Definition diskreter konfokaler Koordinaten, wieder unter Verwendung der genannten neuen diskreten Orthogonalitätsbedingung. Diese Definition führt zu einer größeren Klasse von Netzen als im ersten Ansatz und beinhaltet beliebege Umparametriesierungen. Es wird gezeigt, dass diese diskreten konfokalen Koordinaten durch eine äquivalente geometrische Konstruktion durch Polarität in einer Folge von klassischen konfokalen Quadriken charakterisiert ist. Verschiedene Folgen entsprechen verschiedenen diskreten Parametrisierungen. Wir geben eine Vielzahl von konkreten Beispielen an, insebesondere eine Parametrisierung durch Jacobi elliptische Funktionen. Ein besonderes Beispiel von diskreten konfokalen Koordinaten im zwei-dimensionalen Fall ist durch Inkreisnetze gegeben. Inkreisnetzte sind durch zwei Folgen von Geraden in der Ebene mit der Kombinatorik des Quadratgitters gegeben, so dass alle elementaren Vierecke einen Inkreis besitzen. Wir klassifizieren und integrieren die zugehörige Laguerre-geometrische Verallgemeinerung der Schachbrettinkreisnetze. Weitere Aspekte der neuen diskreten Orthogonalitätzbedingung werden in der Einleitung behandelt. Unter anderem diskrete Lamé-Koeffizienten, diskrete Fokalnetze, diskrete Parallelnetze, sowie der Zusammenhang zu Paaren von zirkulären und konischen Netzen.DFG, 195170736, Discretization in geometry and dynamic

On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs

Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics and systems of confocal quadrics. Intimate connections with classical deformations of quadrics that are isometric along asymptotic lines and circular cross-sections of quadrics are revealed. The existence of octahedral webs of surfaces of Blaschke type generated by asymptotic and characteristic lines that are diagonally related to lines of curvature is proved theoretically and established constructively. Appropriate samplings (grids) of these webs lead to three-dimensional extensions of non-planar IC nets. Three-dimensional octahedral grids composed of planes and spatially extending (checkerboard) IC-nets are shown to arise in connection with systems of confocal quadrics in Minkowski space. In this context, the Laguerre geometric notion of conical octahedral grids of planes is introduced. The latter generalise the octahedral grids derived from systems of confocal quadrics in Minkowski space. An explicit construction of conical octahedral grids is presented. The results are accompanied by various illustrations which are based on the explicit formulae provided by the theory