arXiv subjects

E. Tsukerman

Publications and source records attributed to E. Tsukerman.

3 recordsLinked to original sources

arXiv2016

Iterating evolutes and involutes

We study iterations of two classical constructions, the evolutes and involutes of plane curves, and we describe the limiting behavior of both constructions on a class of smooth curves with singularities given by their support functions. Next we study two kinds of discretizations of these constructions: the curves are replaced by polygons, and the evolutes are formed by the circumcenters of the triples of consecutive vertices, or by the incenters of the triples of consecutive sides. The space of polygons is a vector bundle over the space of the side directions, and both kinds of evolutes define vector bundle morphisms. In both cases, we describe the linear maps of the fibers. In the first case, the induced map of the base is periodic, whereas, in the second case, it is an averaging transformation. We also study the dynamics of the related inverse constructions, the involutes of polygons. In addition to the theoretical study, we performed numerous computer experiments; some of the observations remain unexplained.

math.DS↗

arXiv2014

Remarks on the the circumcenter of mass

Suppose that to every non-degenerate simplex Delta in n-dimensional Euclidean space a `center' C(Delta) is assigned so that the following assumptions hold: (i) The map that assigns C(Delta) to Delta commutes with similarities and is invariant under the permutations of the vertices of the simplex; (ii) The map that assigns Vol(Delta) C(Delta) to Delta is polynomial in the coordinates of the vertices of the simplex. Then C(Delta) is an affine combination of the center of mass and the circumcenter of Delta (with the coefficients independent of the simplex). The motivation for this theorem comes from the recent study of the circumcenter of mass of simplicial polytopes by the authors and by A. Akopyan.

math.MG↗

arXiv2013

Polygonal Bicycle Paths and the Darboux Transformation

A Bicycle $(n,k)$-gon is an equilateral $n$-gon whose $k$ diagonals are of equal length. In this paper we consider periodic bicycle $(n,k)$-paths, which are a natural variation in which the polygon is replaced with a periodic polygonal path.

math.DS↗