AI 中文总结
受Descartes相切圆链问题启发，研究施泰纳4链半径可行性问题，利用施泰纳链数值不变量刻画符合条件的正有序四元组，给出算法解决方案及相关示例
AI 中文摘要
受Descartes关于相切圆链的著名问题的解决方案（由[此处URL]和[此处URL]给出）的启发，我们针对[此处URL]所研究的施泰纳4链半径的可行性问题给出了一种算法解决方案。更确切地说，我们获得了与构成施泰纳4链的圆的曲率四元组相符的正有序四元组的完整刻画。我们的方法必不可少地利用了[此处URL]和[此处URL]近期论文中引入的施泰纳链的数值不变量。我们计算了施泰纳4链的所有不变量，并确立了这些不变量之间的代数关系，这些关系被用于主要结果的证明。还给出了若干相关观察结果和说明性示例。关键词：施泰纳链、母圆、施泰纳 porism、poristic 施泰纳链、Descartes 圆定理、曲率的不变矩、不变量之间的代数关系、对称施泰纳链、施泰纳链的可行性问题
英文摘要
Motivated by the solution of Descartes famous problem on the chains of tangent circles given by D.Mathews and O.Zymaris, we give an algorithmic solution to the feasibility problem for radii of Steiner 4-chains studied by K.Kirajiev. More precisely, we obtain a complete characterization of the ordered quadruples of positive numbers which coincide with a quadruple of curvatures of the circles forming a Steiner 4-chain. Our approach makes essential use of the numerical invariants of Steiner chains introduced in a recent paper of R.Schwartz and S.Tabachnikov. We compute all invariants of Steiner 4-chains and establish the algebraic relations between those invariants which are used in the proof of the main result. Several related observations and illustrative examples are also given. Keywords: Steiner chain, parent circles, Steiner porism, poristic Steiner chains, Descartes circle theorem, invariant moments of curvatures, algebraic relations between invariants, symmetric Steiner chains, feasibility problem for Steiner chains
