Complex invariants of poristic Steiner 4-chains

Abstract

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

In this version substantially revises the previous one. We've changed the title, the invariant moments are reformulated in terms of complex moments, new algebraic relations are derived and the application is shifted from feasibility of radii to feasibility of centers of Steiner 4-chains. Some new results are added in the section 3 and 4. The exposition and references have been updated