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索迪圆

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SoddyCircles

给定三个非共线点，构造三个相切圆，使得每个点为一个圆的圆心，且这些圆两两相切。然后，恰好存在两个不相交的圆，它们与所有这三个圆相切。这些圆被称为内索迪圆和外索迪圆，它们的圆心分别被称为内索迪中心 S 和外索迪中心 S^'。

弗雷德里克·索迪 (Frederick Soddy) (1936) 给出了计算索迪圆 (r_4) 半径的公式，已知其他三个圆的半径 r_i (i=1, 2, 3)。关系式是

2(epsilon_1^2+epsilon_2^2+epsilon_3^2+epsilon_4^2)=(epsilon_1+epsilon_2+epsilon_3+epsilon_4)^2,
(1)

其中 epsilon_i=+/-kappa_i=+/-1/r_i 是所谓的弯曲率，定义为圆的有符号曲率。如果所有接触都是外切的，则符号都取正号，而如果一个圆包围了其他三个圆，则该圆的符号取负号 (Coxeter 1969)。使用二次公式求解 epsilon_4，用半径而不是曲率表示，并简化得到

r_4^+/-=(r_1r_2r_3)/(r_1r_2+r_1r_3+r_2r_3+/-2sqrt(r_1r_2r_3(r_1+r_2+r_3))).
(2)

这里，负解对应于外索迪圆，正解对应于内索迪圆。

任意四个两两相切的圆，其切点中，相对切点的连线三线共点，“相对”在这里指的是确定一个切点的两个圆与确定另一个切点的两个圆不同 (Eppstein 2001)。这一事实引出了第一和第二 Eppstein 点。

这个公式被称为笛卡尔圆定理，因为它在笛卡尔时代就已为人所知。索迪将结果扩展到相切球体，而 Gosper 进一步将结果扩展到 n+2 个两两相切的 n 维超球体。

Bellew 推导出了一个更广泛的公式，适用于一个圆被 n 个圆包围，而这 n 个圆又被另一个圆外接的情况。关系式是

[n(c_n-1)^2+1]sum_(i=1)^(n+1)kappa_i^2+n(3nc_n^2-2n-6)c_n^2(c_n-1)^2=[(f(n))/(n(c_n-1)+1)]^2,
(3)

其中 kappa_(n+1) 是中心圆的曲率，

f(n)=[n(c_n-1)^2+1]sum_(i=1)^(n+1)kappa_i+nc_n(c_n-1)[nc_n^2+(3-n)c_n-4]
(4)

并且

c_n=csc(pi/n).
(5)

对于 n=3，这简化为笛卡尔圆定理

2sum_(i=1)^4kappa_i^2=(sum_(i=1)^4kappa_i)^2.
(6)

另请参阅

Apollonian Gasket, Apollonius Circle, Apollonius' Problem, Arbelos, Bend, Bowl of Integers, Circumcircle, Descartes Circle Theorem, Excentral Triangle, Four Coins Problem, Hart's Theorem, Inner Soddy Center, Inner Soddy Circle, Malfatti Circles, Outer Soddy Center, Outer Soddy Circle, Pappus Chain, Soddy Centers, Soddy Triangles, Sphere Packing, Steiner Chain, Tangent Circles, Tangent Spheres

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参考文献

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索迪圆

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Weisstein, Eric W. “索迪圆.” 来自 MathWorld--Wolfram Web 资源。 https://mathworld.net.cn/SoddyCircles.html

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