Invariants and areas of Steiner 4-chains

TL;DR

This paper investigates invariants of Steiner 4-chains, deriving formulas for curvature invariants, and applies these to develop an algorithmic criterion and identify chains with extremal areas, extending recent research.

Contribution

It introduces new formulas for invariants of Steiner 4-chains and applies them to criteria and extremal area identification, generalizing prior results.

Findings

Derived invariant moments of curvatures in Steiner 4-chains.

Presented an algorithmic criterion for feasibility of Steiner 4-chains.

Identified Steiner chains with extremal areas.

Abstract

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.