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Invariants and areas of Steiner 4-chains

Guna Bibileishvili, Ana Diakvnishvili

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.

Invariants and areas of Steiner 4-chains

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.
Paper Structure (5 sections, 8 theorems, 54 equations)

This paper contains 5 sections, 8 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. Invariants of poristic Steiner $4$-chains
  3. Feasibility criterion for the radii of Steiner $4$-chains
  4. Extremal problems for poristic Steiner $4$-chains
  5. Concluding remarks

Key Result

Proposition 1

For a given pair of Soddy circles with gauge $(R, r, d)$, the minimal and maximal possible values of poristic radii $r_i$ are while the minimal and maximal values of poristic curvatures are respectively. For any $r \in [r_*, r^*],$ the poristic family contains a circle of radius $r$.

Theorems & Definitions (17)

  • Remark 1
  • Proposition 1
  • Proposition 2
  • Corollary 1
  • Remark 2
  • Lemma 1
  • Lemma 2
  • Theorem 1
  • Remark 3
  • Theorem 2
  • ...and 7 more