Complex invariants of poristic Steiner 4-chains

TL;DR

This paper investigates the complex invariants of Steiner 4-chains, establishing algebraic relations and invariance properties, with applications to the centers' feasibility problem, extending previous work on Steiner chains.

Contribution

It introduces the computation of invariant complex moments for poristic Steiner 4-chains and derives algebraic relations, extending invariance results to these chains and related configurations.

Findings

Computed invariant complex moments for poristic Steiner 4-chains.

Established algebraic relations between these invariants.

Extended results to Steiner 3-chains and applied to centers' feasibility.

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