arXiv subjects

Ana Diakvnishvili

Publications and source records attributed to Ana Diakvnishvili.

4 recordsLinked to original sources

arXiv2026

Feasibility problem for the radii of Steiner 4-chains

Motivated by the solution of Descartes famous problem on the chains of tangent circles given by D.Mathews and O.Zymaris, we give an algorithmic solution to the feasibility problem for radii of Steiner 4-chains studied by K.Kirajiev. More precisely, we obtain a complete characterization of the ordered quadruples of positive numbers which coincide with a quadruple of curvatures of the circles forming a Steiner 4-chain. Our approach makes essential use of the numerical invariants of Steiner chains introduced in a recent paper of R.Schwartz and S.Tabachnikov. We compute all invariants of Steiner 4-chains and establish the algebraic relations between those invariants which are used in the proof of the main result. Several related observations and illustrative examples are also given. Keywords: Steiner chain, parent circles, Steiner porism, poristic Steiner chains, Descartes circle theorem, invariant moments of curvatures, algebraic relations between invariants, symmetric Steiner chains, feasibility problem for Steiner chains

math.MG↗

arXiv2025

Complex invariants of poristic Steiner 4-chains

We are concerned with the Steiner chains consisting of four circles. More precisely, we deal with the so-called complex moments of Steiner 4-chains introduced in a recent paper by J.Lagarias, C.Mallows and A.Wilks. We compute the invariant complex moments of poristic Steiner 4-chains and establish certain algebraic relations between those invariants. To this end we use the invariance of certain moments of curvatures of poristic Steiner chains established by R.Schwartz and S.Tabachnikov, combined with the computation of these moments for the so-called symmetric Steiner 4-chains. We also present analogous results for poristic Steiner 3-chains and give an application to the feasibility problem for the centers of Steiner 4-chains. KEYWORDS: Steiner chain, parent circles, Steiner porism, poristic Steiner chains, Descartes circle theorem, invariant bending moments, complex moments of Steiner chains, algebraic relations between invariants

math.GM↗

arXiv2025

Bicentric configurations of pentagonal linkages

Let $P$ be a planar $n$-gon with the sidelengths $a_1, \ldots, a_n$ and let us denote by $L=L(P)$ the corresponding planar polygonal linkage. We are concerned with the problem of finding conditions on the sidelengths $a_i$ which guarantee the existence of a bicentric configuration of $L$. Specifically, we present some related results on tangential and chordal polygons and characterize pentagonal linkages having bicentric configurations.

math.AG↗

arXiv2025

Invariants and areas of Steiner 4-chains

We are concerned with the invariants of Steiner chains consisting of four circles. In particular, we compute the invariant moments of curvatures in Steiner 4-chains and give two applications of the obtained formulas. Specifically, we present an algorithmic feasibility criterion for Steiner 4-chains and identify the poristic Steiner chains having extremal areas, which yields a generalisation of the main results of a recent paper by K.Kiradjiev. The proofs are based on the invariants of Steiner chains described by R.Schwarz and S.Tabachnikov and on the relations between the radii of neighbouring poristic circles established by P.Yiu.

math.CO↗