Related papers: Discrete Koenigs nets, inscribed quadrics and auto…
This paper studies the discrete differential geometry of the checkerboard pattern inscribed in a quadrilateral net by connecting edge midpoints. It turns out to be a versatile tool which allows us to consistently define principal nets,…
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…
In the projective plane, we consider congruences of straight lines with the combinatorics of the square grid and with all elementary quadrilaterals possessing touching inscribed conics. The inscribed conics of two combinatorially…
We investigate the common underlying discrete structures for various smooth and discrete nets. The main idea is to impose the characteristic properties of the nets not only on elementary quadrilaterals but also on larger parameter…
We provide a convincing discretisation of Demoulin's $\Omega$-surfaces along with their specialisations to Guichard and isothermic surfaces with no loss of integrable structure.
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…
We introduce the Koenigs lattice, which is a new integrable reduction of the quadrilateral lattice (discrete conjugate net) and provides natural integrable discrete analogue of the Koenigs net. We construct the Darboux-type transformations…
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…
We investigate some characteristic properties of specific Weingarten surfaces in the three-dimensional Euclidean space using the nets of the lines of curvature resp. the asymptotic lines on both central surfaces of them.
Conjugate line parametrizations of surfaces were first discretized almost a century ago as quad meshes with planar faces. With the recent development of discrete differential geometry, two discretizations of principal curvature line…
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…
We consider general integrable curve nets in Euclidean space as a particular integrable geometry invariant with respect to rigid motions and net-preserving reparameterisations. For the purpose of their description, we first give an overview…
It is given the diffeomorphism classification on generic singularities of tangent varieties to curves with arbitrary codimension in a projective space. The generic classifications are performed in terms of certain geometric structures and…
Recent techniques have been successful in reconstructing surfaces as level sets of learned functions (such as signed distance fields) parameterized by deep neural networks. Many of these methods, however, learn only closed surfaces and are…
We give an elaborated treatment of discrete isothermic surfaces and their analogs in different geometries (projective, M\"obius, Laguerre, Lie). We find the core of the theory to be a novel projective characterization of discrete isothermic…
A novel class of discrete integrable surfaces is recorded. This class of discrete O surfaces is shown to include discrete analogues of classical surfaces such as isothermic, `linear' Weingarten, Guichard and Petot surfaces. Moreover,…
The goal of this study is to provide a method for computing the following: Given a network of curves in 3d (satisfying a condition at the intersection points), compute efficiently a smooth surface such that the curves are geodesics on it.…
In the present paper, we propose a new discrete surface theory on 3-valent embedded graphs in the 3-dimensional Euclidean space which are not necessarily discretization or approximation of smooth surfaces. The Gauss curvature and the mean…
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…
We introduce the dual Koenigs lattices, which are the integrable discrete analogues of conjugate nets with equal tangential invariants, and we find the corresponding reduction of the fundamental transformation. We also introduce the notion…