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

💡 Research Summary

The paper investigates invariant complex moments of Steiner chains consisting of four circles (Steiner 4‑chains) in the classical Steiner porism setting. After recalling the basic definitions, the authors introduce the “gauge” (R, r, d) of a pair of parent circles: R and r are the radii of the outer and inner fixed circles, and d is the distance between their centers. The poristic condition for an n‑chain is the quadratic relation d² = (R − r)² − 4qRr with q = tan²(π/n). For a given chain G they define the curvature (b‑ends) a = 1/r, A = −1/R and the k‑th bending moment Iₖ(G)=∑bⱼᵏ. It is known (Schwartz‑Tabachnikov) that for a fixed pair of parent circles the first n − 1 bending moments I₁,…,I_{n−1} are invariant throughout the poristic family; the paper records explicit formulas for n = 3 and n = 4 (equations (3)–(7)).

The central novelty is the introduction of complex moments
 J_{k,m}(G)=∑ b_i^{k} z_i^{m},
where z_i are the complex coordinates of the circle centers. For 0 ≤ m ≤ k ≤ n−1 these quantities are also invariant (Lagarias‑Mallows‑Wilks). The authors observe that, after placing the outer parent circle at the origin and aligning the axis of porism with the real axis, all invariant complex moments become real numbers, revealing a hidden real structure.

To compute these invariants the paper exploits symmetric Steiner chains. Two types of symmetry are distinguished: axial (the chain contains the maximal and minimal radius circles and is symmetric with respect to the axis of porism) and lateral (a chain symmetric with respect to the axis but touching it at two points). Using the quadratic relation (8) for the curvatures of neighboring circles, the authors solve explicitly for the radii and curvatures of the axial chain C^{**} and the lateral chain when n = 4 (q = 1). The coefficients α, β, γ in (8) simplify to (13), allowing closed‑form expressions for the curvatures b⁺, b⁻ of the lateral chain. With these data the invariant complex moments J_{1,1}, J_{2,1}, J_{2,2} are obtained in elementary algebraic form.

A major contribution is the derivation of algebraic relations among the invariant moments. For example, for n = 4 the authors prove a polynomial identity involving I₁, I₂, I₃ and the complex moments (e.g., I₁·J_{2,2} − I₂² + … = 0). Such relations were absent from earlier works (