420 citations · 1.3k across the 61 of their papers we have counts for
35 papers · 1 filter
Constant mean curvature surfaces from ring patterns: Geometry from combinatorics
Alexander I. Bobenko, Tim Hoffmann, Nina Smeenk
We define discrete constant mean curvature (cmc) surfaces in the three-dimensional Euclidean and Lorentz spaces in terms of sphere packings with orthogonally intersecting circles.…
Minimal reflection surfaces in Combinatorics of curvature lines and minimal surfaces based on fundamental pentagons
Alexander I. Bobenko, Sebastian Heller, Nicolas Schmitt
We study compact minimal surfaces in the 3-sphere which are constructed by successive reflections from a minimal -gon -- so-called minimal reflection surfaces. The minimal -g…
Isothermic tori with one family of planar curvature lines and area constrained hyperbolic elastica
Alexander I. Bobenko, Tim Hoffmann, Andrew O. Sageman-Furnas
In 1883, Darboux gave a local classification of isothermic surfaces with one family of planar curvature lines using complex analytic methods. His choice of real reduction cannot co…
Circular Nets with Spherical Parameter Lines and Terminating Laplace Sequences
Alexander I. Bobenko, Alexander Y. Fairley
The focus is on circular nets with one or two families of spherical parameter lines, which are treated in Möbius geometry. These circular nets provide a discretisation of surfaces…
Compact Bonnet Pairs: isometric tori with the same curvatures
Alexander I. Bobenko, Tim Hoffmann, Andrew O. Sageman-Furnas
We explicitly construct a pair of immersed tori in three dimensional Euclidean space that are related by a mean curvature preserving isometry. These Bonnet pair tori are the first…
Constant mean curvature surfaces based on fundamental quadrilaterals
Alexander I. Bobenko, Sebastian Heller, Nicholas Schmitt
We describe the construction of CMC surfaces with symmetries in and using a CMC quadrilateral in a fundamental tetrahedron of a tessellation of the spac…
On mutually diagonal nets on (confocal) quadrics and 3-dimensional webs
Arseniy V. Akopyan, Alexander I. Bobenko, Wolfgang K. Schief +1
Canonical parametrisations of classical confocal coordinate systems are introduced and exploited to construct non-planar analogues of incircular (IC) nets on individual quadrics an…
Minimal n-Noids in hyperbolic and anti-de Sitter 3-space
Alexander I. Bobenko, Sebastian Heller, Nicholas Schmitt
We construct minimal surfaces in hyperbolic and anti-de Sitter 3-space with the topology of a -punctured sphere by loop group factorization methods. The end behavior of the surf…
Checkerboard incircular nets. Laguerre geometry and parametrisation
Alexander I. Bobenko, Wolfgang K. Schief, Jan Techter
We present a procedure which allows one to integrate explicitly the class of checkerboard IC-nets which has recently been introduced as a generalisation of incircular (IC) nets. Th…
Multi-Nets. Classification of discrete and smooth surfaces with characteristic properties on arbitrary parameter rectangles
Alexander I. Bobenko, Helmut Pottmann, Thilo Rörig
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 elem…
On a discretization of confocal quadrics. II. A geometric approach to general parametrizations
Alexander I. Bobenko, Wolfgang K. Schief, Yuri B. Suris +1
We propose a discretization of classical confocal coordinates. It is based on a novel characterization thereof as factorizable orthogonal coordinate systems. Our geometric discreti…
Discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. A discrete Lawson correspondence
Alexander I. Bobenko, Pascal Romon
The main result of this paper is a discrete Lawson correspondence between discrete CMC surfaces in R^3 and discrete minimal surfaces in S^3. This is a correspondence between two di…
On a discretization of confocal quadrics. I. An integrable systems approach
Alexander I. Bobenko, Wolfgang K. Schief, Yuri B. Suris +1
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 pr…
Circle complexes and the discrete CKP equation
Alexander I. Bobenko, Wolfgang K. Schief
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 syst…
On a new conformal functional for simplicial surfaces
Alexander I. Bobenko, Martin P. Weidner
We introduce a smooth quadratic conformal functional and its weighted version where is the extrinsic intersection…
Supercyclidic nets
Alexander I. Bobenko, Emanuel Huhnen-Venedey, Thilo Rörig
Supercyclides are surfaces with a characteristic conjugate parametrization consisting of two families of conics. Patches of supercyclides can be adapted to a Q-net (a discrete quad…
Discrete line complexes and integrable evolution of minors
Alexander I. Bobenko, Wolfgang K. Schief
Based on the classical Plücker correspondence, we present algebraic and geometric properties of discrete integrable line complexes in . Algebraically, these are encoded in a…
Discrete constant mean curvature nets in space forms: Steiner's formula and Christoffel duality
A. Bobenko, U. Hertrich-Jeromin, I. Lukyanenko
We show that the discrete principal nets in quadrics of constant curvature that have constant mixed area mean curvature can be characterized by the existence of a Königs dual in a…
Conformal Structures and Period Matrices of Polyhedral Surfaces
Alexander I. Bobenko, Christian Mercat, Markus Schmies
We recall the theory of linear discrete Riemann surfaces and show how to use it in order to interpret a surface embedded in R^3 as a discrete Riemann surface and compute its basis…
A curvature theory for discrete surfaces based on mesh parallelity
Alexander I. Bobenko, Helmut Pottmann, Johannes Wallner
We consider a general theory of curvatures of discrete surfaces equipped with edgewise parallel Gauss images, and where mean and Gaussian curvatures of faces are derived from the f…
Integrable discrete nets in Grassmannians
Vsevolod E. Adler, Alexander I. Bobenko, Yuri B. Suris
We consider discrete nets in Grassmannians which generalize Q-nets (maps with planar elementary quadrilaterals) and Darboux nets ($\m…
Discrete Koenigs nets and discrete isothermic surfaces
Alexander I. Bobenko, Yuri B. Suris
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 quadrilateral…
Surfaces from Circles
Alexander I. Bobenko
In the search for appropriate discretizations of surface theory it is crucial to preserve such fundamental properties of surfaces as their invariance with respect to transformation…
Curvature line parametrization from circle patterns
A. I. Bobenko, S. P. Tsarev
We study local and global approximations of smooth nets of curvature lines and smooth conjugate nets by respective discrete nets (circular nets and planar quadrilateral nets) with…