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Gaiane Panina

Publications and source records attributed to Gaiane Panina.

51 records · Page 3Linked to original sources
arXiv2016

Motion planning and control of a planar polygonal linkage

For a polygonal linkage, we produce a fast navigation algorithm on its configuration space. The basic idea is to approximate the configuration space by the vertex-edge graph of its cell decomposition discovered by the first author. The algorithm has three aspects: (1) the number of navigation steps does not exceed 15 (independent of the linkage), (2) each step is a disguised flex of a quadrilateral from one triangular configuration to another, which is a well understood type of flex, and (3) each step can be performed explicitly by adding some extra bars and obtaining a mechanism with one degree of freedom.

math.MG↗
arXiv2016

Cyclopermutohedron: geometry and topology

The face poset of the permutohedron realizes the combinatorics of linearly ordered partitions of the set $[n]=\{1,...,n\}$. Similarly, the cyclopermutohedron is a virtual polytope that realizes the combinatorics of cyclically ordered partitions of the set $[n+1]$. The cyclopermutohedron was introduced by the third author by motivations coming from configuration spaces of polygonal linkages. In the paper we prove two facts: (1) the volume of the cyclopermutohedron equals zero, and (2) the homology groups $H_k$ for $k=0,...,n-2$ of the face poset of the cyclopermutohedron are non-zero free abelian groups. We also present a short formula for their ranks.

math.MG↗
arXiv2015

Equilibria of point charges on convex curves

We study the equilibrium positions of three points on a convex curve under influence of the Coulomb potential. We identify these positions as orthotripods, three points on the curve having concurrent normals. This relates the equilibrium positions to the caustic (evolute) of the curve. The concurrent normals can only meet in the core of the caustic, which is contained in the interior of the caustic. Moreover, we give a geometric condition for three points in equilibrium with positive charges only. For the ellipse we show that the space of orthotripods is homeomorphic to a 2-dimensional bounded cylinder.

math.DG↗
arXiv2015

Point Charges and Polygonal Linkages

We investigate the critical points of Coulomb potential of point charges placed at the vertices of a planar polygonal linkage. It is shown that, for a collection of positive charges on a pentagonal linkage, there is a unique critical point in the set of convex configurations which is the point of absolute minimum. This enables us to prove that two controlling charges are sufficient to navigate between any two convex configurations of a pentagonal linkage.

math.MG↗
arXiv2015

Volume and lattice points counting for the cyclopermutohedron

The face lattice of the permutohedron realizes the combinatorics of linearly ordered partitions of the set $[n]=\{1,...,n\}$. Similarly, the cyclopermutohedron is a virtual polytope that realizes the combinatorics of cyclically ordered partitions of $[n]$. It is known that the volume of the standard permutohedron equals the number of trees with $n$ labeled vertices multiplied by $\sqrt{n}$. The number of integer points of the standard permutohedron equals the number of forests on $n$ labeled vertices. In the paper we prove that the volume of the cyclopermutohedron also equals some weighted number of forests, which eventually reduces to zero. We also derive a combinatorial formula for the number of integer points in the cyclopermutohedron. Another object of the paper is the configuration space of a polygonal linkage $L$. It has a cell decomposition $\mathcal{K}(L)$ related to the face lattice of cyclopermutohedron. Using this relationship, we introduce and compute the volume $Vol(\mathcal{K}(L))$.

math.MG↗
arXiv2014

Cyclopermutohedron

It is known that the $k$-faces of the permutohedron $Π_n$ are labeled by (all possible) linearly ordered partitions of the set $[n]=\{1,...,n\}$ into $(n-k)$ non-empty parts. The incidence relation corresponds to the refinement: a face $F$ contains a face $F'$ whenever the label of $F'$ refines the label of $F$. In the paper we consider the cell complex ${CP}$ defined in analogous way, replacing linear ordering by cyclic ordering. Namely, $k$-cells of the complex ${CP}$ are labeled by (all possible) cyclically ordered partitions of the set $[n+1]=\{1,...,n, n+1\}$ into $(n+1-k)$ non-empty parts, where $(n+1-k)>2$. The incidence relation again corresponds to the refinement: a cell $F$ contains a cell $F'$ whenever the label of $F'$ refines the label of $F$. In particular, two vertices are joined by an edge whenever their labels differ on a permutation of two neighbor elements. The complex ${CP}$ cannot be represented by a convex polytope, since it is not a combinatorial sphere (not even a combinatorial manifold). However, it can be represented by some \textit{virtual polytope} (Minkowski difference of two convex polytopes) which we call "cyclopermutohedron" $\mathcal{CP}_{n+1}$. It is defined explicitly, as a weighted Minkowski sum of line segments. Informally, the cyclopermutohedron can be viewed as "permutohedron with diagonals". One of the motivations is that the cyclopermutohedron is a "universal" polytope for moduli spaces of polygonal linkages.

math.MG↗
arXiv2014

Simple game induced manifolds

Starting by a simple game $Q $ as a combinatorial data, we build up a cell complex $M(Q)$, whose construction resembles combinatorics of the permutohedron. The cell complex proves to be a combinatorial manifold; we call it the \textit{ simple game induced manifold.} By some motivations coming from polygonal linkages, we think of $Q$ and of $M(Q)$ as of\textit{ a quasilinkage} and the \textit{moduli space of the quasilinkage} respectively. We present some examples of quasilinkages and show that the moduli space retains many properties of moduli space of polygonal linkages. In particular, we show that the moduli space $M(Q)$ is homeomorphic to the space of stable point configurations on $S^1$, for an associated with a quasilinkage notion of stability.

math.GT↗
arXiv2013

Coulomb control of polygonal linkages

Equilibria of polygonal linkage with respect to Coulomb potential of point charges placed at the vertices of linkage are considered. It is proved that any convex configuration of a quadrilateral linkage is the point of global minimum of Coulomb potential for appropriate values of charges of vertices. Similar problems are treated for the equilateral pentagonal linkage. Some corollaries and applications in the spirit of control theory are also presented.

math.MG↗
arXiv2012

Extremal polygons in R^3

The oriented area function $A$ is (generically) a Morse function on the space of planar configurations of a polygonal linkage. We are lucky to have an easy description of its critical points as cyclic polygons and a simple formula for the Morse index of a critical point. However, for planar polygons, the function $A$ in many cases is not a perfect Morse function. In particular, for an equilateral pentagonal linkage it has one extra local maximum (except for the global maximum) and one extra local minimum. In the present paper we consider the space of 3D configurations of a polygonal linkage. For an appropriate generalization $S$ of the area function $A$ the situation becomes nicer: we again have an easy description of critical points and a simple formula for the Morse index. In particular, unlike the planar case, for an equilateral linkage with odd number of edges the function $S$ is always a perfect Morse function and fits the lacunary principle. Therefore cyclic equilateral polygons can be interpreted as independent generators of the homology groups of the (decorated) configuration space.

math.GT↗
arXiv2011

Swap action on moduli spaces of polygonal linkages

The basic object of the paper is the moduli space $M_{2,3}(L)$ of a closed polygonal linkage either in $\mathbb{R}^2$ or in $\mathbb{R}^3$. As was originally suggested by G. Khimshiashvili, the space $M_{2}(L)$ is equipped with the oriented area function $A$, whereas (as is suggested in the paper) $M_{3}(L)$ is equipped with the vector area function $S$. The latter are generically Morse functions, whose critical points have a nice description. In the preprint, we define a \textit{swap action} (that is, the action of some group generated by edge transpositions) on the space $M_{2,3}(L)$ which preserves the functions $A$ and $S$ and the Morse points. We prove that the commutant of the group acts trivially, present some computer experiments and formulate a conjecture.

math.MG↗
arXiv2011

Around a conjecture by R. Connelly, E. Demaine, and G. Rote

Denote by $M(P)$ the configuration space of a planar polygonal linkage, that is, the space of all possible planar configurations modulo congruences, including configurations with self-intersections. A particular interest attracts its subset $M^o(P) \subset M(P)$ of all configurations \emph{without} self-intersections. R. Connelly, E. Demaine, and G. Rote proved that $M^o(P)$ is contractible and conjectured that so is its closure $\bar{M^o(P)}$. We disprove this conjecture by showing that a special choice of $P$ makes the homologies $H_k(\bar{M^o(P)})$ non-trivial.

math.AT↗
arXiv2010

Morse index of a cyclic polygon

It is known that cyclic configurations of a planar polygonal linkage are critical points of the signed area function. In the paper, we announce an explicit formula of the Morse index for the signed area of a cyclic configuration. It depends not only on the combinatorics of a cyclic configuration, but also includes some metric characterization.

math.MG↗
arXiv2010

Flattening single-vertex origami: the non-expansive case

A single-vertex origami is a piece of paper with straight-line rays called creases emanating from a fold vertex placed in its interior or on its boundary. The Single-Vertex Origami Flattening problem asks whether it is always possible to reconfigure the creased paper from any configuration compatible with the metric, to a flat, non-overlapping position, in such a way that the paper is not torn, stretched and, for rigid origami, not bent anywhere except along the given creases. Streinu and Whiteley showed how to reduce the problem to the carpenter's rule problem for spherical polygons. Using spherical expansive motions, they solved the cases of open < πand closed <= 2πspherical polygons. Here, we solve the case of open polygons with total length between [π, 2π), which requires non-expansive motions. Our motion planning algorithm works in a finite number of discrete steps, for which we give precise bounds depending on both the number of links and the angle deficit.

cs.CG↗
arXiv2006

Planar pseudo-triangulations, spherical pseudo-tilings and hyperbolic virtual polytopes

We wish to draw attention to an interesting and promising interaction of two theories. On the one hand, it is the theory of \textbf{pseudo-triangulations} which was useful for implicit solution of thecarpenter's rule problem and proved later to give a nice tool for graph embeddings. On the other hand, it is the theory of hyperbolic virtual polytopes which arose from an old uniqueness conjecture for convex bodies (A. D. Alexandrov's problem): suppose that a constant $C$ separates (non-strictly) everywhere the principal curvature radii of a smooth 3-dimensional convex body $K$. Then $K$ is necessarily a ball of radius $C$. The two key ideas are: Passing from planar pseudo-triangulations to spherical pseudo-tilings, we avoid non-poited vertices. Instead, we use pseudo-di-gons. A theorem on spherically embedded Laman-plus-one graphs is announced. The difficult problem of hyperbolic polytopes constructing can be reduced to finding spherically embedded graphs.

math.MG↗