Minimal surfaces from infinitesimal deformations of circle packings

Lam, Wai Yeung

doi:10.1016/j.aim.2019.106939

Cited by 13 publications

(12 citation statements)

References 21 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It is referred to the fixed boundary curve of a surface area that is minimal with respect to other surfaces with the same boundary. This is equivalent to having zero mean curvature [11][12][13].…”

Section: Introductionmentioning

confidence: 99%

Smarandache Ruled Surfaces according to Frenet-Serret Frame of a Regular Curve in E 3

Ouarab

2021

Abstract and Applied Analysis

View full text Add to dashboard Cite

In this paper, we introduce original definitions of Smarandache ruled surfaces according to Frenet-Serret frame of a curve in E 3 . It concerns TN-Smarandache ruled surface, TB-Smarandache ruled surface, and NB-Smarandache ruled surface. We investigate theorems that give necessary and sufficient conditions for those special ruled surfaces to be developable and minimal. Furthermore, we present examples with illustrations.

show abstract

Section: Introductionmentioning

confidence: 99%

Smarandache Ruled Surfaces according to Frenet-Serret Frame of a Regular Curve in E 3

Ouarab

2021

Abstract and Applied Analysis

View full text Add to dashboard Cite

show abstract

“…Interestingly, the main theorem of [27] says that a planar embedded framework has an edgelength equilibrium stress if and only if a triangulation of its dual medial graph has an equilibrium stress satisfying the formally different condition j ω i j p i − p j 2 = 0 at each vertex. This condition is connected to orthogonal circle patterns [27] and discrete minimal surfaces [28]. ♦…”

Section: Definition 22mentioning

confidence: 99%

“…Remark 2.3. W. Lam (2018, personal communication) has shown that edge-length equilibrium stresses are equivalent to the holomorphic quadratic differentials of Köbe type from [27]. In particular, for planar frameworks without boundary (in the sense of [27]), there are none.…”

Section: The Packing Manifoldmentioning

confidence: 99%

Rigidity for sticky discs

2019

View full text Add to dashboard Cite

We study the combinatorial and rigidity properties of disk packings with generic radii. We show that a packing of n disks in the plane with generic radii cannot have more than 2n − 3 pairs of disks in contact.The allowed motions of a packing preserve the disjointness of the disk interiors and tangency between pairs already in contact (modeling a collection of sticky disks). We show that if a packing has generic radii, then the allowed motions are all rigid body motions if and only if the packing has exactly 2n − 3 contacts. Our approach is to study the space of packings with a fixed contact graph. The main technical step is to show that this space is a smooth manifold, which is done via a connection to the Cauchy-Alexandrov stress lemma.Our methods also apply to jamming problems, in which contacts are allowed to break during a motion. We give a simple proof of a finite variant of a recent result of Connelly, et al.[13] on the number of contacts in a jammed packing of disks with generic radii. Sticky disk and framework rigidity A motion of a packing, called a flex, is one that preserves the radii, the disjointness of the disk interiors, as well as tangency between pairs of disks with a corresponding edge in the contact graph. (The last condition makes the disks "sticky".) A packing is rigid when all its flexes arise from rigid body motions; otherwise, it is flexible.Since any packing has a neighborhood on which the contact graph remains fixed along any flex (one must move at least some distance before a new contact can appear), the constraints on the packing are locally equivalent to preserving the pairwise distances between the circle centers. Forgetting that the disks have radii and must remain disjoint and keeping only the distance contraints between the centers, we get exactly "framework rigidity", where we have a configuration * Figure 1: A rigid sticky disk packing and its underlying bar framework. Disks are the green circles with red center points / joints, the bars are blue segments. p = (p 1 , . . . , p n ) of n points in a d-dimensional Euclidean space and a graph G; the pair (G, p) is called a (bar-and-joint) framework. The allowed flexes of the points are those that preserve the distances between the pairs indexed by the edges of G. As with packings, a framework is rigid when all its flexes arise from rigid body motions and otherwise flexible. Given a packing (p, r) with contact graph G, we call the framework (G, p) its underlying framework.Motivations Rigidity of sticky disk packings, and the relationship to frameworks, has several related, but formally different motivations. The first comes from the study of colloidal matter [31], which is made of micrometer-sized particles that interact with "short range potentials" [2, 33] that, in a limit, behave like sticky disks [22] (spheres in 3d).Secondly, there is a connection to jammed packings. Here one has a "container", which can shrink uniformly and push the disks together. In this setting, we do not require any contact graph to be preserved. A fundamen...

show abstract

“…ey refer to the fixed boundary curve of a surface area that is minimal with respect to other surfaces with the same boundary. ey are characterized with vanishing mean curvature [6][7][8].…”

Section: Introductionmentioning

confidence: 99%

$NC$ -Smarandache Ruled Surface and $NW$ -Smarandache Ruled Surface according to Alternative Moving Frame in $E$

Ouarab

2021

Journal of Mathematics

View full text Add to dashboard Cite

In this study, original definitions of NC − Smarandache ruled surface and NW − Smarandache ruled surface according to the alternative moving frame are introduced in E 3 . The main results of the study are presented in theorems that give necessary and sufficient conditions for those special surfaces to be developable and minimal. Finally, an example with illustrations is presented.

show abstract

Minimal surfaces from infinitesimal deformations of circle packings

Cited by 13 publications

References 21 publications

Smarandache Ruled Surfaces according to Frenet-Serret Frame of a Regular Curve in E 3

Smarandache Ruled Surfaces according to Frenet-Serret Frame of a Regular Curve in E 3

Rigidity for sticky discs

$NC$ -Smarandache Ruled Surface and $NW$ -Smarandache Ruled Surface according to Alternative Moving Frame in $E$

Contact Info

Product

Resources

About

Minimal surfaces from infinitesimal deformations of circle packings

Cited by 13 publications

References 21 publications

Smarandache Ruled Surfaces according to Frenet-Serret Frame of a Regular Curve in E 3

Smarandache Ruled Surfaces according to Frenet-Serret Frame of a Regular Curve in E 3

Rigidity for sticky discs

NC -Smarandache Ruled Surface and NW -Smarandache Ruled Surface according to Alternative Moving Frame in E

Contact Info

Product

Resources

About

$NC$ -Smarandache Ruled Surface and $NW$ -Smarandache Ruled Surface according to Alternative Moving Frame in $E$