On Weingarten Transformations of Hyperbolic Nets

Huhnen-Venedey, Emanuel; Schief, W. K.

doi:10.1093/imrn/rnt354

Cited by 5 publications

(3 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The present work is a continuation of our previous works on the discretizations of smooth orthogonal and asymptotic nets by cyclidic and hyperbolic nets, respectively [5,20,21]. We will now review the corresponding theory for cyclidic nets and illustrate the core concepts that may serve as a structural guideline for this article.…”

Section: Introductionmentioning

confidence: 93%

Supercyclidic Nets

Bobenko

Huhnen-Venedey

Rörig

2016

Int Math Res Notices

View full text Add to dashboard Cite

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 quadrilateral net with planar faces) such that neighboring surface patches share tangent planes along common boundary curves. We call the resulting patchworks "supercyclidic nets" and show that every Q-net in RP 3 can be extended to a supercyclidic net. The construction is governed by a multidimensionally consistent 3D system. One essential aspect of the theory is the extension of a given Q-net in RP N to a system of circumscribed discrete torsal line systems. We present a description of the latter in terms of projective reflections that generalizes the systems of orthogonal reflections which govern the extension of circular nets to cyclidic nets by means of Dupin cyclide patches.2010 Mathematics Subject Classification. 51A05, 53A20, 37K25, 65D17.

show abstract

Section: Introductionmentioning

confidence: 93%

Supercyclidic Nets

Bobenko

Huhnen-Venedey

Rörig

2016

Int Math Res Notices

View full text Add to dashboard Cite

show abstract

“…Hyperbolic nets are remarkable structures in discrete differential geometry. Huhnen-Venedey and Schief [49] use them for the study of discrete Weingarten transformations, and there is ongoing research by Schief who employs them in a discrete theory of projective minimal surfaces.…”

Section: Smooth Negatively Curved Surfaces From Ruled Patchesmentioning

confidence: 99%

Architectural geometry

Pottmann

Eigensatz²,

Vaxman

et al. 2015

Computers & Graphics

237

153

View full text Add to dashboard Cite

a b s t r a c tAround 2005 it became apparent in the geometry processing community that freeform architecture contains many problems of a geometric nature to be solved, and many opportunities for optimization which however require geometric understanding. This area of research, which has been called architectural geometry, meanwhile contains a great wealth of individual contributions which are relevant in various fields. For mathematicians, the relation to discrete differential geometry is significant, in particular the integrable system viewpoint. Besides, new application contexts have become available for quite some old-established concepts. Regarding graphics and geometry processing, architectural geometry yields interesting new questions but also new objects, e.g. replacing meshes by other combinatorial arrangements. Numerical optimization plays a major role but in itself would be powerless without geometric understanding. Summing up, architectural geometry has become a rewarding field of study. We here survey the main directions which have been pursued, we show real projects where geometric considerations have played a role, and we outline open problems which we think are significant for the future development of both theory and practice of architectural geometry.

show abstract

“…Hence, we introduce the notion of discrete envelopes of lattice Lie quadrics and show that any discrete asymptotic net associated with the even Demoulin sublattice may be regarded as an envelope of the lattice Lie quadrics of the corresponding discrete asymptotic net associated with the odd Demoulin sublattice and vice versa. Here, we exploit the theory of hyperbolic nets developed in detail in [6,7].…”

Section: Introductionmentioning

confidence: 99%

On the Combinatorics of Demoulin Transforms and (Discrete) Projective Minimal Surfaces

McCarthy

Schief

2016

Discrete Comput Geom

View full text Add to dashboard Cite

The classical Demoulin transformation is examined in the context of discrete differential geometry. We show that iterative application of the Demoulin transformation to a seed projective minimal surface generates a Z 2 lattice of projective minimal surfaces. Known and novel geometric properties of these Demoulin lattices are discussed and used to motivate the notion of lattice Lie quadrics and associated discrete envelopes and the definition of the class of discrete projective minimal and Q-surfaces (PMQ-surfaces). We demonstrate that the even and odd Demoulin sublattices encode a two-parameter family of pairs of discrete PMQ-surfaces with the property that one discrete PMQsurface constitute an envelope of the lattice Lie quadrics associated with the other.

show abstract

On Weingarten Transformations of Hyperbolic Nets

Cited by 5 publications

References 16 publications

Supercyclidic Nets

Supercyclidic Nets

Architectural geometry

On the Combinatorics of Demoulin Transforms and (Discrete) Projective Minimal Surfaces

Contact Info

Product

Resources

About