And yet it moves: Paradoxically moving linkages in kinematics

Schicho, Josef

doi:10.1090/bull/1721

Cited by 7 publications

(2 citation statements)

References 36 publications

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“…There exist minimally rigid graphs such that for a carefully chosen edge length assignment, a vertex in this graph traces out a trajectory, while fixing some edge into place. We refer to [22] for an overview of the classification of such paradoxically moving graphs.…”

Section: State Of the Art: Coupler Curvesmentioning

confidence: 99%

Coupler curves of moving graphs and counting realizations of rigid graphs

Grasegger¹,

Hilany²,

Lubbes³

2022

Preprint

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A calligraph is a graph that for almost all edge length assignments moves with one degree of freedom in the plane, if we fix an edge and consider the vertices as revolute joints. The trajectory of a distinguished vertex of the calligraph is called its coupler curve. To each calligraph we uniquely assign a vector consisting of three integers. This vector bounds the degrees and geometric genera of irreducible components of the coupler curve. A graph, that up to rotations and translations admits finitely many, but at least two, realizations into the plane for almost all edge length assignments, is a union of two calligraphs. We show that this number of realizations is equal to a certain inner product of the vectors associated to these two calligraphs. As an application we obtain an improved algorithm for counting numbers of realizations, and by counting realizations we characterize invariants of coupler curves.

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Section: State Of the Art: Coupler Curvesmentioning

confidence: 99%

Coupler curves of moving graphs and counting realizations of rigid graphs

Grasegger¹,

Hilany²,

Lubbes³

2022

Preprint

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“…embeddings (called, realizations) or has infinitely-many of them. In the latter case, it is called paradoxically moving [26].…”

Section: Introductionmentioning

confidence: 99%

The tropical discriminant of a polynomial map on a plane

Hilany¹

2022

Preprint

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The discriminant of a polynomial map between two spaces is an object central to many problems in mathematics. Standard methods for computing it rely on elimination techniques which can be inefficient. This paper concerns polynomial maps on the two-dimensional torus defined over the field of Puiseux series with complex coefficients. We present a combinatorial procedure for computing the tropical curve of the discriminant of such maps that satisfy some mild genericity conditions. Thanks to the advances made in tropical geometry, our results give rise to a new tool for investigating the discriminant of a complex polynomial map on the plane, and for computing its Newton polytope.

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Computing the Non-properness Set of Real Polynomial Maps in the Plane

El Hilany,

Tsigaridas

2023

Vietnam J. Math.

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We introduce novel mathematical and computational tools to develop a complete algorithm for computing the set of non-properness of polynomials maps in the plane. In particular, this set, which we call the Jelonek set, is a subset of $$\mathbb {K}^2$$ K 2 , where a dominant polynomial map $$f: \mathbb {K}^2 \rightarrow \mathbb {K}^2$$ f : K 2 → K 2 is not proper; $$\mathbb {K}$$ K could be either $$\mathbb {C}$$ C or $$\mathbb {R}$$ R . Unlike all the previously known approaches we make no assumptions on f whenever $$\mathbb {K} = \mathbb {R}$$ K = R ; this is the first algorithm with this property. The algorithm takes into account the Newton polytopes of the polynomials. As a byproduct we provide a finer representation of the set of non-properness as a union of semi-algebraic curves, that correspond to edges of the Newton polytopes, which is of independent interest. Finally, we present a precise Boolean complexity analysis of the algorithm and a prototype implementation in maple.

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And yet it moves: Paradoxically moving linkages in kinematics

Cited by 7 publications

References 36 publications

Coupler curves of moving graphs and counting realizations of rigid graphs

Coupler curves of moving graphs and counting realizations of rigid graphs

The tropical discriminant of a polynomial map on a plane

Computing the Non-properness Set of Real Polynomial Maps in the Plane

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