A new field of discrete differential geometry is presently emerging on the border between differential and discrete geometry. Whereas classical differential geometry investigates smooth geometric shapes (such as surfaces), and discrete geometry studies geometric shapes with finite number of elements (such as polyhedra)…
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48 results for “discrete differential geometry”
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 invar…
In this paper we introduce a discrete integrable system generalizing the discrete (real) cross-ratio system in to complex values of a generalized cross-ratio by considering as a real section of the complex Plücker quadric, realized as the space of two-spheres in We develop the geometry of the Plücker…
New representations for discrete surfaces derived from dual transforms.
problem Constructing discrete surfaces in differential geometry.
method Using -dual transform and lightlike Gauss maps in Laguerre geometry.
result All discrete linear Weingarten surfaces arise via Weierstrass-type representations.
Discretization of curves is an ancient topic. Even discretization of curves with an eye toward differential geometry is over a century old. However there is no general theory or methodology in the literature, despite the ubiquitous use of discrete curves in mathematics and science. There are conflicting definitions of …
Discrete geometry model approximates Willmore energy.
problem Approximating the Willmore energy for triangulated surfaces.
method A discrete energy defined in the spirit of discrete differential geometry converges to the Willmore energy.
result The discrete energy converges to the Willmore energy in the sense of -convergence.
Defines discrete differential geometry concepts in homotopy type theory.
problem No existing definition of Euler characteristic for comparison.
method Type families on higher inductive types, simplicial complexes, principal bundles, connections, curvature, vector fields, index.
result Theorem relating total curvature and total index, key to proving Gauss-Bonnet and Poincaré-Hopf theorems.
Fractured Meshesmath.NA
Generalized meshes for non-regular geometries, including fractures.
problem Discretization of partial differential equations in non-regular geometries.
method Introduces generalized meshes with overlapping elements and flexible adjacency relations.
result Discrete differential forms on virtually inflated meshes characterize the trace space of forms in surrounding volumes.
Skew parallelogram nets factorize, encompassing discrete differential geometry.
problem Factorization of polynomials in discrete differential geometry.
method Lax representation, Bäcklund transformations, factorization of polynomials.
result Skew parallelogram nets encompass all systems with polynomial representations.
In the framework of nonassociative geometry (hep-th/0003238) a unified description of continuum and discrete spacetime is proposed. In our approach at the Planck scales the spacetime is described as a so-called "diodular discrete structure" which at large spacetime scales `looks like' a differentiable manifold. After a…
We present the first steps of a procedure which discretises surface theory in classical projective differential geometry in such a manner that underlying integrable structure is preserved. We propose a canonical frame in terms of which the associated projective Gauss-Weingarten and Gauss-Mainardi-Codazzi equations adop…
We prove that many aspects of the differential geometry of embedded Riemannian manifolds can be formulated in terms of multi linear algebraic structures on the space of smooth functions. In particular, we find algebraic expressions for Weingarten's formula, the Ricci curvature and the Codazzi-Mainardi equations. For ma…
Develops TCD maps to relate discrete differential geometry and cluster algebras.
problem Capturing constraints and dynamics in discrete differential geometry.
method Triple crossing diagram maps (TCD maps) and geometric operations.
result Establishes a hierarchy of cluster structures on TCD maps.
Study of discrete Koenigs nets and their properties.
problem Characterization and properties of discrete Koenigs nets.
method Generalization of inscribed conics to inscribed quadrics and study of Koenigs d-grids.
result Established a bijection between Koenigs d-grids and pairs of discrete autoconjugate curves.
Weingarten transformations which, by definition, preserve the asymptotic lines on smooth surfaces have been studied extensively in classical differential geometry and also play an important role in connection with the modern geometric theory of integrable systems. Their natural discrete analogues have been investigated…
The purpose of this thesis is to study classical combinatorial objects, such as polytopes, polytopal complexes, and subspace arrangements, using tools that have been developed in combinatorial topology, especially those tools developed in connection with (discrete) differential geometry, geometric group theory and low-…
Develops discrete geometry for non-constant curvature surfaces.
problem Modeling surfaces of non-constant curvature, especially with non-constant negative curvature.
method Derived and numerically integrated Lelieuvre formulas for hyperbolic surfaces. Proposed iterative and fast marching methods for solving implicit equations and computing geodesic distances.
result Explicit construction of immersions is not provided, but equations are described implicitly.
Discrete curvaturemath.DG
Survey on discrete curvature concepts for polygons and polyhedral surfaces.
problem Defining curvature for discrete structures like polygons and polyhedral surfaces.
method Explains curvature notions for polygons, polyhedral surfaces, and abstract polyhedral manifolds.
result Discrete curvature theorems parallel classical theorems in differential geometry.
Develops combinatorial theory of vector bundles on simplicial complexes.
problem Creating a discrete theory for vector bundles and connections on simplicial complexes.
method Introduces discrete exterior covariant derivative and applies it to various geometric objects.
result Flat discrete connections yield a cochain complex computing twisted de Rham cohomology.
A Novel Mathematical Model of Protein Interactions from the Perspective of Electron Delocalizationq-bio.BM
A new model explains protein interactions via electron delocalization.
problem Understanding how protein interactions affect each other.
method Quantized discrete differential geometry of n-simplices.
result Allosteric regulation follows from the model of interactions.
New finite element method for complex forms in any dimension.
problem Discretization of complex forms in arbitrary dimensions.
method Finite element discretization of -form-valued -forms on triangulations.
result Generalizes existing finite element methods for various tensor fields.
New discrepancy function compares discrete probability measures considering space geometry.
problem Comparing discrete probability measures in a geometrically meaningful way.
method Proposes the Fourier Discrepancy Function, proving convexity, differentiability, and providing gradient formula.
result Proves the Fourier Discrepancy is convex, twice differentiable, and provides an explicit gradient formula.
The paper concerns discrete versions of the three well-known results of projective differential geometry: the four vertex theorem, the six affine vertex theorem and the Ghys theorem on four zeroes of the Schwarzian derivative. We study geometry of closed polygonal lines in $\bbRP^d$ and prove that polygons satisfying a…
In this paper higher order mimetic discretizations are introduced which are firmly rooted in the geometry in which the variables are defined. The paper shows how basic constructs in differential geometry have a discrete counterpart in algebraic topology. Generic maps which switch between the continuous differential for…
Principal binetsmath-ph
New discretizations of principal curvature lines discovered.
problem Discretizing principal curvature line parametrizations.
method Generalization of polar pairs of line congruences in the Lie quadric.
result New discretizations of orthogonal and Gauss-orthogonal parametrizations.
Asymptotic net is an important concept in discrete differential geometry. In this paper, we show that we can associate affine discrete geometric concepts to an arbitrary non-degenerate asymptotic net. These concepts include discrete affine area, mean curvature, normal and co-normal vector fields and cubic form, and the…
Group lattices (Cayley digraphs) of a discrete group are in natural correspondence with differential calculi on the group. On such a differential calculus geometric structures can be introduced following general recipes of noncommutative differential geometry. Despite of the non-commutativity between functions and (gen…
Finding appropriate notions of discrete holomorphic maps and, more generally, conformal immersions of discrete Riemann surfaces into 3-space is an important problem of discrete differential geometry and computer visualization. We propose an approach to discrete conformality that is based on the concept of holomorphic l…
Unified framework for various geometric constructions.
problem Organizing diverse geometric constructions.
method Introducing TCD maps and defining local moves.
result Two distinct cluster structures on TCD maps.
Geometric approach uses Bäcklund transformations to create integrable discrete analogs of surface nets.
problem Creating integrable discrete analogs of surface nets and conjugate nets.
method Interpreting classical differential geometry results through Bäcklund transformations and applying permutability properties.
result Integrable discrete analogs of asymptotic and conjugate nets are constructed.
Within a framework of noncommutative geometry, we develop an analogue of (pseudo) Riemannian geometry on finite and discrete sets. On a finite set, there is a counterpart of the continuum metric tensor with a simple geometric interpretation. The latter is based on a correspondence between first order differential calcu…
The paper explores discrete isothermic nets using checkerboard patterns in quadrilateral nets.
problem Defining and understanding discrete isothermic nets in quadrilateral nets.
method Using checkerboard patterns and discrete differential geometry to define and analyze isothermic nets.
result The class of isothermic nets is invariant under dualization and Moebius transformations.
On the Resolution of Partial Differential Equations for Lattice Structures on Smooth Manifoldsmath.AP
This paper solves PDEs for embedding discrete lattices into smooth manifolds.
problem Embedding discrete lattices into smooth manifolds while preserving geometric and topological properties.
method Rigorous mathematical framework and analysis of partial differential equations (PDEs).
result Existence and regularity of solutions to PDEs under initial boundary conditions.
Hamilton's Ricci flow (RF) equations were recently expressed in terms of a sparsely-coupled system of autonomous first-order nonlinear differential equations for the edge lengths of a d-dimensional piecewise linear (PL) simplicial geometry. More recently, this system of discrete Ricci flow (DRF) equations was further s…
We study a discrete dynamical system designed to find a 'most holomorphic' connection on a smooth complex vector bundle . We examine the relation between the distance of the chern classes of from the axis of the Hodge diamond and singularity formation. Canonical connections and canonical metrics pulled b…
Discrete vector bundles are important in Physics and recently found remarkable applications in Computer Graphics. This article approaches discrete bundles from the viewpoint of Discrete Differential Geometry, including a complete classification of discrete vector bundles over finite simplicial complexes. In particular,…
A primer on Script Geometrymath.DG
This text is an exposition of a new approach into discrete differential geometry, called Script Geometry. In difference to classic approaches while scripts are based on complexes of cells we are not limited to simplicial complexes. One of the principal concepts of Script Geometry is the notion of tightness which is a m…
Discrete conjugate systems are quadrilateral nets with all planar faces. Discrete orthogonal systems are defined by the additional property of all faces being concircular. Their geometric properties allow one to consider them as proper discretization of conjugate, resp. orthogonal coordinate systems of classical differ…
A novel symmetry in nanocarbons: pre-constant discrete principal curvature structurecond-mat.mtrl-sci
Novel symmetry found in nanocarbons' discrete principal curvature structure.
problem Identifying novel symmetries in nanocarbons' geometric structures.
method First-principles calculations and discrete geometry analysis.
result Discovery of a novel symmetry (pre-constant discrete principal curvature) in nanocarbons.
Constructs the moduli stack of elliptic curves as an orbifold.
problem Moduli stack construction for elliptic curves.
method Analytic orbifold construction, non-effective actions.
result Provides a self-contained account for young researchers.
This is an unrefereed lecture note based on lectures in 'Introductory Workshop on Discrete Differential Geometry' at Korea University on January 21--24, 2019. In this note, we discuss topological crystallography, which is a mathematical theory of crystal structures. The most symmetric structure among all placements of …
Sheaves on graphs link to noncommutative geometry.
problem Exploring noncommutative geometry concepts on graphs.
method Sheaf theory and simplicial sets.
result Enhanced understanding of discrete noncommutative geometry.
Novel bistable structures made from four-bar linkages, proving existence and construction.
problem Existence and construction of bistable mechanical structures composed of four-bar linkages.
method Geometric construction starting from infinitesimally flexible quad nets, applying Whiteley de-averaging.
result Construction of bistable structures from well-known quad nets, allowing control of geometric parameters.
Generalized differential geometry uses infinitesimals to solve singularities in differential equations.
problem Understanding differential equations with singularities and nonlinearities.
method Introducing infinitesimals and infinities to handle singularities and nonlinearities rigorously.
result A Riemannian manifold can be embedded into a generalized manifold where singularities vanish and products of nonlinearities make sense.
Classically, isothermic surfaces are characterized as those surfaces which are "divisible into infinitesimal squares by their curvature lines". This characterization is the direct analogue to the definition of discrete isothermic nets. In order to understand the relations between the discrete and the smooth theory bett…
Neural GDEs improve graph prediction by blending discrete structures and differential equations.
problem Dynamic graph prediction challenges in irregularly sampled data.
method Continuous-depth graph neural networks (GNNs) with Neural GDEs.
result Neural GDEs enhance performance across various applications.
Polyhedra can mimic constant curvature surfaces, even with self-intersections.
problem Understanding curvature constraints in discrete vs. smooth settings.
method Constructive proof showing any surface can be realized as a polyhedral surface with uniform angular defect.
result Closed surfaces can be realized as polyhedral surfaces with constant angular defect.
This work develops discrete Gaussian models for vector-valued data on triangular meshes.
problem Discrete representation of continuous vector-valued environmental data.
method Develops discrete intrinsic Gaussian processes for vector-valued data on triangular meshes using discrete differential operators.
result Models can capture harmonic flows, incorporate boundary conditions, and model non-stationary data.