Related papers: Discrete channel surfaces

We prove that all discrete isothermic nets with a family of planar or spherical lines of curvature can be obtained from special discrete holomorphic maps via lifted-folding. This novel approach is a generalization and discretization of a…

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…

We introduce a new general-purpose approach to deep learning on 3D surfaces, based on the insight that a simple diffusion layer is highly effective for spatial communication. The resulting networks are automatically robust to changes in…

Scale-discretised wavelets yield a directional wavelet framework on the sphere where a signal can be probed not only in scale and position but also in orientation. Furthermore, a signal can be synthesised from its wavelet coefficients…

A discrete harmonic surface is a trivalent graph which satisfies the balancing condition in the 3-dimensional Euclidean space and achieves energy minimizing under local deformations. Given a topological trivalent graph, a holomorphic…

We define discrete flat surfaces in hyperbolic 3-space from the perspective of discrete integrable systems and prove properties that justify the definition. We show how these surfaces correspond to previously defined discrete constant mean…

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…

Polyhedral surfaces are fundamental objects in architectural geometry and industrial design. Whereas closeness of a given mesh to a smooth reference surface and its suitability for numerical simulations were already studied extensively, the…

We propose a notion of discrete elastic and area-constrained elastic curves in 2-dimensional space forms. Our definition extends the well-known discrete Euclidean curvature equation to space forms and reflects various geometric properties…

On the basis of loop group decompositions (Birkhoff decompositions), we give a discrete version of the nonlinear d'Alembert formula, a method of separation of variables of difference equations, for discrete constant negative Gauss curvature…

Differential calculus on discrete sets is developed in the spirit of noncommutative geometry. Any differential algebra on a discrete set can be regarded as a `reduction' of the `universal differential algebra' and this allows a systematic…

In this paper, we extend the notion of Schwarz reflection principle for smooth minimal surfaces to the discrete analogues for minimal surfaces, and use it to create global examples of discrete minimal nets with high degree of symmetry.

We study a new discretization of the Gaussian curvature for polyhedral surfaces. This discrete Gaussian curvature is defined on each conical singularity of a polyhedral surface as the quotient of the angle defect and the area of the Voronoi…

The conditions for a cuspidal edge, swallowtail and other fundamental singularities are given in the context of Lie sphere geometry. We then use these conditions to study the Lie sphere transformations of a surface.

We define a new theory of discrete Riemann surfaces and present its basic results. The key idea is to consider not only a cellular decomposition of a surface, but the union with its dual. Discrete holomorphy is defined by a straightforward…

We present an overview of the differential geometry of curves and surfaces using examples from soft matter as illustrations. The presentation requires a background only in vector calculus and is otherwise self-contained.

We investigate geometric properties of surfaces given by certain formulae. In particular, we calculate the singular curvature and the limiting normal curvature of such surfaces along the set of singular points consisting of singular points…

We consider smooth isotropic immersions from the 2-dimensional torus into $R^{2n}$, for $n \geq 2$. When $n = 2$ the image of such map is an immersed Lagrangian torus of $R^4$. We prove that such isotropic immersions can be approximated by…

We extend two of the methods previously introduced to find discrete symmetries of differential equations to the case of difference and differential-difference equations. As an example of the application of the methods, we construct the…

Motivated by the theory of quantum waveguides, we investigate the spectrum of the Laplacian, subject to Dirichlet boundary conditions, in a curved strip of constant width that is defined as a tubular neighbourhood of an infinite curve in a…