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### On dimensional rigidity of bar-and-joint frameworks

Abdo Y. Alfakih
2007 Discrete Applied Mathematics
Then an r-configuration p together with graph G, where adjacent vertices of G are constrained to stay the same distance apart, is called a bar-and-joint framework (or a framework) in R r , and is denoted  ...  In this paper we introduce the notion of dimensional rigidity of frameworks, and we study the problem of determining whether or not a given G(p) is dimensionally rigid.  ...  Acknowledgment We would like to thank the referee for his/her comments and suggestions.  ...

### Bracing Zonohedra With Special Faces

Gyula Nagy
2015 YBL Journal of Built Environment
This paper deals with the flexibility of this kind of mechanisms, and investigates the rigidity of the braced framework.  ...  A necessary and sufficient condition for the rigidity of the braced rhombic face zonohedra is posed.  ...  INFINITESIMAL RIGIDITY OF THE FRAMEWORK Rigidity and infinitesimal rigidity of bar-and-joint frameworks The position of the joints X and Y are denoted by p(x), p(y).  ...

### The Rigidity of Frameworks: Theory and Applications

Jessica Sidman, Audrey St. John
2017 Notices of the American Mathematical Society
Because characterizing 3-dimensional bar-and-joint rigidity remains an open challenge, researchers turn to a different model of rigidity called the body-and-bar model.  ...  A body-and-bar framework is composed of rigid bodies, constrained by bars placed between pairs of bodies and attached at universal joints.  ...

### Infinitesimally Rigid Polyhedra. I. Statics of Frameworks

Walter Whiteley
1984 Transactions of the American Mathematical Society
In this paper the dual (and equivalent) concept of static rigidity for frameworks is used to describe the behavior of bar and joint frameworks built around convex (and other) polyhedra.  ...  The techniques are also applied to derive the static rigidity of tensegrity frameworks (with cables and struts in place of bars), and the static rigidity of frameworks projectively equivalent to known  ...  Statics of bar and joint frameworks. We begin with the promised introduction to the static rigidity of bar and joint frameworks.  ...

### Infinitesimally rigid polyhedra. I. Statics of frameworks

Walter Whiteley
1984 Transactions of the American Mathematical Society
In this paper the dual (and equivalent) concept of static rigidity for frameworks is used to describe the behavior of bar and joint frameworks built around convex (and other) polyhedra.  ...  The techniques are also applied to derive the static rigidity of tensegrity frameworks (with cables and struts in place of bars), and the static rigidity of frameworks projectively equivalent to known  ...  Statics of bar and joint frameworks. We begin with the promised introduction to the static rigidity of bar and joint frameworks.  ...

### Generic rigidity in three-dimensional bond-bending networks

D J Jacobs
1998 Journal of Physics A: Mathematical and General
2, and after it is argued in section 3 that Laman's theorem can be generalized to bar-joint frameworks having no implied-hinge joints.  ...  Based on the proposition that an all subgraph constraint counting characterization of generic rigidity is recovered in three-dimensional bar-joint networks having no implied hinge joints, an efficient  ...  Acknowledgments I thank M F Thorpe for many in-depth discussions about generic rigidity and its relevance to glass networks and thank B Hendrickson for making many valuable suggestions.  ...

### Create a Rigid and Safe Grid-Like Structure

Attila Kovács, Gyula Nagy Kem
2019 Periodica polytechnica. Civil engineering
We offer some conclusions, including perspectives and future developments in the rigidity of scaffolds and tall building as symmetrical and grid-like bar-joint frameworks.  ...  In our consideration, we study those relationships that based on our frame using their finite element analyses and some new result in optimizations of structural design.  ...  We offer some conclusions, including perspectives and future developments in the rigidity of scaffolds and tall building as bar-joint frameworks.  ...

### The rigidity of periodic body-bar frameworks on the three-dimensional fixed torus

E. Ross
2013 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
We present necessary and sufficient conditions for the generic rigidity of body-bar frameworks on the three-dimensional fixed torus.  ...  These frameworks correspond to infinite periodic body-bar frameworks in R^3 with a fixed periodic lattice.  ...  The majority of this research was carried out at the Fields Institute for Research in Mathematical Sciences in Toronto, Canada.  ...

### Point-hyperplane frameworks, slider joints, and rigidity preserving transformations

Yaser Eftekhari, Bill Jackson, Anthony Nixon, Bernd Schulze, Shin-ichi Tanigawa, Walter Whiteley
2018 Journal of combinatorial theory. Series B (Print)
A one-to-one correspondence between the infinitesimal motions of bar-joint frameworks in R d and those in S d is a classical observation by Pogorelov, and further connections among different rigidity models  ...  This enables us to understand correspondences between point-hyperplane rigidity, classical bar-joint rigidity, and scene analysis.  ...  We also thank HIM (Bonn) and ICMS (Edinburgh) for hosting further rigidity workshops in October 2015 and May 2016, respectively, and we thank DIMACS (Rutgers) for hosting the workshop on 'Distance Geometry  ...

### On Rigidity of Unit-Bar Frameworks

József Solymosi, Ethan White
2019 Graphs and Combinatorics
A framework in Euclidean space consists of a set of points called joints, and line segments connecting pairs of joints called bars.  ...  One example is a motion of a single joint, where all other joints are unmoving, such that the movement of the one joint is perpendicular to all bars attached to it.  ...  The rigidity matrix of G(p) has a row for each bar, and d columns for each joint, corresponding to each coordinate of the joint.  ...

### Page 4398 of Mathematical Reviews Vol. , Issue 93h [page]

1993 Mathematical Reviews
Then, given any subset of n joints of the original framework (not lying in an (n — 2)-dimensional hyperplane), a rigid unit-bar framework can be created so that some subset of n joints has the same lengths  ...  Such a unit-bar framework is said to be flexible if the points (or joints) can be continuously moved, preserving the lengths of the unit bars, so that at least one pair of joints change their mutual distance  ...

### Maxwell-Laman counts for bar-joint frameworks in normed spaces [article]

Derek Kitson, Bernd Schulze
2014 arXiv   pre-print
In this paper we generalize this tool and introduce a rigidity matrix for bar-joint frameworks in arbitrary finite dimensional real normed vector spaces.  ...  The rigidity matrix is a fundamental tool for studying the infinitesimal rigidity properties of Euclidean bar-joint frameworks.  ...  In this paper, we take a more general viewpoint and consider the infinitesimal rigidity properties of bar-joint frameworks in an arbitrary finite dimensional real normed vector space (also referred to  ...

### Infinitesimal rigidity for non-Euclidean bar-joint frameworks

D. Kitson, S. C. Power
2014 Bulletin of the London Mathematical Society
The minimal infinitesimal rigidity of bar-joint frameworks in the non-Euclidean spaces (R^2, ||.||_q) are characterised in terms of (2,2)-tight graphs.  ...  Specifically, a generically placed bar-joint framework (G,p) in the plane is minimally infinitesimally rigid with respect to a non-Euclidean l^q norm if and only if the underlying graph G = (V,E) contains  ...  For the theory of bar-joint frameworks in finite dimensional Euclidean space see, for example, Asimow and Roth [1] , [2] , Gluck [4] and Graver, Servatius and Servatius [5] .  ...

### Symmetry-extended counting rules for periodic frameworks

S. D. Guest, P. W. Fowler
2013 Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
P.W.F. thanks the Royal Society and Leverhulme Trust for a Senior Research Fellowship.  ...  We thank Holger Mitschke (Universität Erlangen-Nürnberg) for helpful discussions during the drafting of this paper.  ...  Examples for bar-and-joint frameworks (a) A two-dimensional bar-and-joint framework: the kagome framework The plane group of the kagome framework is p6m, and the point group isomorphic to the factor group  ...

### When is a symmetric pin-jointed framework isostatic? [article]

Robert Connelly, Patrick Fowler, Simon Guest, Bernd Schulze, Walter Whiteley
2008 arXiv   pre-print
In particular, it turns out that an isostatic framework in 2D can belong to one of only six point groups.  ...  This paper shows how, for an isostatic framework, these conditions imply very simply stated restrictions on the numbers of those structural components that are unshifted by the symmetry operations of the  ...  These conditions included some over-all counts on bars and joints, along with subcounts on special classes of bars and joints (bars on mirrors or perpendicular to mirrors, bars centered on the axis of  ...
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