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35
First Steps in Tropical Geometry
 CONTEMPORARY MATHEMATICS
"... Tropical algebraic geometry is the geometry of the tropical semiring (R, min, +). Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete descr ..."
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Cited by 81 (11 self)
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Tropical algebraic geometry is the geometry of the tropical semiring (R, min, +). Its objects are polyhedral cell complexes which behave like complex algebraic varieties. We give an introduction to this theory, with an emphasis on plane curves and linear spaces. New results include a complete description of the families of quadrics through four points in the tropical projective plane and a counterexample to the incidence version of Pappus’ Theorem.
Straightening polygonal arcs and convexifying polygonal cycles
 DISCRETE & COMPUTATIONAL GEOMETRY
, 2000
"... Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles ..."
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Cited by 78 (30 self)
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Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewisedifferentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular, this result settles the wellstudied carpenter’s rule conjecture.
Secondorder rigidity and prestress stability for tensegrity frameworks
 SIAM J. Discrete Math
, 1996
"... Abstract. This paper defines two concepts of rigidity for tensegrity frameworks (frameworks with cables, bars, and struts): prestress stability and secondorder rigidity. We demonstrate a hierarchy of rigidityfirstorder rigidity implies prestress stability implies secondorder rigidity implies ri ..."
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Cited by 41 (10 self)
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Abstract. This paper defines two concepts of rigidity for tensegrity frameworks (frameworks with cables, bars, and struts): prestress stability and secondorder rigidity. We demonstrate a hierarchy of rigidityfirstorder rigidity implies prestress stability implies secondorder rigidity implies rigiditymfor any framework. Examples show that none of these implications are reversible, even for bar frameworks. Other examples illustrate how these results can be used to create rigid tensegrity frameworks. This paper also develops a duality for secondorder rigidity, leading to a test which combines information on the self stresses and the firstorder flexes of a framework to detect secondorder rigidity. Using this test, the following conjecture of Roth is proven: a plane tensegrity framework, in which the vertices and bars form a strictly convex polygon with additional cables across the interior, is rigid if and only if it is firstorder rigid. Key words, tensegrity frameworks, rigid and flexible frameworks, stability of frameworks, static stress, firstorder motion, secondorder motion AMS subject classifications. Primary, 52C25; Secondary, 70B15, 70C20 1. Introduction. A
RIGIDITY AND SCENE ANALYSIS
"... Rigidity and flexibility of frameworks (motions preserving lengths of bars) and scene analysis (liftings from plane polyhedral pictures to spatial polyhedra) are two core examples of a general class of geometric problems: (a) Given a discrete configuration of points,lines,planes,... in Euclidean spa ..."
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Cited by 37 (8 self)
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Rigidity and flexibility of frameworks (motions preserving lengths of bars) and scene analysis (liftings from plane polyhedral pictures to spatial polyhedra) are two core examples of a general class of geometric problems: (a) Given a discrete configuration of points,lines,planes,... in Euclidean space,
Infinitesimally rigid polyhedra. I. Statics of frameworks
 Transactionsofthe American Mathematical Society
, 1984
"... Abstract. From the time of Cauchy, mathematicians have studied the motions of convex polyhedra, with the faces held rigid while changes are allowed in the dihedral angles. In the 1940s Alexandrov proved that, even with additional vertices along the natural edges, and with an arbitrary triangulation ..."
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Cited by 25 (2 self)
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Abstract. From the time of Cauchy, mathematicians have studied the motions of convex polyhedra, with the faces held rigid while changes are allowed in the dihedral angles. In the 1940s Alexandrov proved that, even with additional vertices along the natural edges, and with an arbitrary triangulation of the natural faces on these vertices, such polyhedra are infinitesimally rigid. 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 static techniques introduced provide a new simplified proof of Alexandrov's theorem, as well as an essential extension which characterizes the static properties of frameworks built with more general patterns on the faces, including frameworks with vertices interior to the faces. The static techniques are presented and employed in a pattern appropriate to the extension of an arbitrary statically rigid framework built around any polyhedron (nonconvex, toroidal, etc.). 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 polyhedral frameworks. Finally, as an exercise to give an additional perspective to the results in 3space, detailed analogues of Alexandrov's theorem are presented for convex 4polytopes built as bar and joint frameworks in 4space. 1. Introduction. Over
RIGIDITY AND POLYNOMIAL INVARIANTS OF CONVEX POLYTOPES
, 2004
"... We present an algebraic approach to the classical problem of constructing a simplicial convex polytope given its planar triangulation and lengths of its edges. We introduce polynomial invariants of a polytope and show that they satisfy polynomial relations in terms of squares of edge lengths. We obt ..."
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Cited by 19 (6 self)
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We present an algebraic approach to the classical problem of constructing a simplicial convex polytope given its planar triangulation and lengths of its edges. We introduce polynomial invariants of a polytope and show that they satisfy polynomial relations in terms of squares of edge lengths. We obtain sharp upper and lower bounds on the degree of these polynomial relations. In a special case of regular bipyramid we obtain explicit formulae for some of these relations. We conclude with a proof of Robbins Conjecture [R2] on the degree of generalized Heron polynomials.
Operations on rigid formations of autonomous agents
 COMMUNICATIONS IN INFORMATION AND SYSTEMS
, 2004
"... This paper is concerned with the maintenance of rigid formations of mobile autonomous agents. A key element in all future multiagent systems will be the role of sensor and communication networks as an integral part of coordination. Network topologies are critically important for autonomous systems ..."
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Cited by 17 (8 self)
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This paper is concerned with the maintenance of rigid formations of mobile autonomous agents. A key element in all future multiagent systems will be the role of sensor and communication networks as an integral part of coordination. Network topologies are critically important for autonomous systems involving mobile underwater, ground and air vehicles and for sensor networks. This paper focuses on developing techniques and strategies for the analysis and design of sensor and network topologies required to achieve a rigid formation for cooperative tasks. Energy efficiency and communication bandwidth are critically important in formations of mobile autonomous agents, and hence strategies that make efficient use of power and energy are beneficial. Therefore, we develop topologies for providing sensing and communications with the minimum number of links, and propose methods requiring the minimum number of changes in the set of links in dynamic missions and maneuvers, including agent departure from a rigid formation, splitting a rigid formation and merging rigid subformations. To do this in a systematic manner, it is necessary to develop a framework for modeling agent formations that characterizes the sensing and communication links needed to maintain the formations. The challenge is that a comprehensive theory of such topologies of formations with sensing and communication limitations is in the earliest stage of development. Central to the development of these techniques and strategies will be the use of tools from rigidity theory, and graph theory.
Infinitesimally locked selftouching linkages with applications to locked trees
 Physical Knots: Knotting, Linking, and Folding of Geometric Objects in 3space
, 2002
"... Abstract. Recently there has been much interest in linkages (barandjoint frameworks) that are locked or stuck in the sense that they cannot be moved into some other configuration while preserving the bar lengths and not crossing any bars. We propose a new algorithmic approach for analyzing whether ..."
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Cited by 16 (10 self)
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Abstract. Recently there has been much interest in linkages (barandjoint frameworks) that are locked or stuck in the sense that they cannot be moved into some other configuration while preserving the bar lengths and not crossing any bars. We propose a new algorithmic approach for analyzing whether planar linkages are locked in many cases of interest. The idea is to examine selftouching or degenerate frameworks in which multiple edges converge to geometrically overlapping configurations. We show how to study whether such frameworks are locked using techniques from rigidity theory, in particular firstorder rigidity and equilibrium stresses. Then we show how to relate locked selftouching frameworks to locked frameworks that closely approximate the selftouching frameworks. Our motivation is that most existing approaches to locked linkages are based on approximations to selftouching frameworks. In particular, we show that a previously proposed locked tree in the plane [BDD + 02] can be easily proved locked using our techniques, instead of the tedious arguments required by standard analysis. We also present a new
Folding and Unfolding
 in Computational Geometry. 2004. Monograph in preparation
, 2001
"... author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Acknowledgments My time as a graduate student has been the best period of my life so far, ..."
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Cited by 16 (3 self)
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author of this thesis. This is a true copy of the thesis, including any required final revisions, as accepted by my examiners. I understand that my thesis may be made electronically available to the public. ii Acknowledgments My time as a graduate student has been the best period of my life so far, and for that wonderful experience I owe many thanks. I had two excellent advisors, Anna Lubiw and Ian Munro. I started working with Anna after I took her two classes on algorithms and computational geometry during my Master’s, which got me excited about both these areas, and even caused me to switch entire fields of computer science, from distributed systems to theory and algorithms. Anna introduced me to Ian when some of our problems in computational geometry turned out to have large data structural components, and my work with Ian blossomed from there. The sets of problems I worked on with Anna and Ian diverged, and both remain my primary interests. Anna and Ian have had a profound influence throughout my academic career. At the most