主題
Search

索迪圓

DOWNLOAD Mathematica Notebook下載 Wolfram 筆記本
SoddyCircles

給定三個非共線點，構造三個相切圓，使得每個點為一個圓的圓心，且這些圓兩兩相切。然後，恰好存在兩個不相交的圓，它們與所有這三個圓相切。這些圓被稱為內索迪圓和外索迪圓，它們的圓心分別被稱為內索迪中心 S 和外索迪中心 S^'。

弗雷德里克·索迪 (Frederick Soddy) (1936) 給出了計算索迪圓 (r_4) 半徑的公式，已知其他三個圓的半徑 r_i (i=1, 2, 3)。關係式是

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 是所謂的彎曲率，定義為圓的有符號曲率。如果所有接觸都是外切的，則符號都取正號，而如果一個圓包圍了其他三個圓，則該圓的符號取負號 (Coxeter 1969)。使用二次公式求解 epsilon_4，用半徑而不是曲率表示，並簡化得到

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)

這裡，負解對應於外索迪圓，正解對應於內索迪圓。

任意四個兩兩相切的圓，其切點中，相對切點的連線三線共點，“相對”在這裡指的是確定一個切點的兩個圓與確定另一個切點的兩個圓不同 (Eppstein 2001)。這一事實引出了第一和第二 Eppstein 點。

這個公式被稱為笛卡爾圓定理，因為它在笛卡爾時代就已為人所知。索迪將結果擴充套件到相切球體，而 Gosper 進一步將結果擴充套件到 n+2 個兩兩相切的 n 維超球體。

Bellew 推匯出了一個更廣泛的公式，適用於一個圓被 n 個圓包圍，而這 n 個圓又被另一個圓外接的情況。關係式是

[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) 是中心圓的曲率，

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)

並且

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

對於 n=3，這簡化為笛卡爾圓定理

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

另請參閱

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

使用 探索

參考文獻

Berger, M.; Pansu, P.; Berry, J.-P.; and Saint-Raymond, X. Problems in Geometry. New York: Springer-Verlag, 1984.Boyd, D. W. "The Sequence of Radii of the Apollonian Packing." Math. Comput. 39, 249-254, 1982.Coxeter, H. S. M. "The Problem of Apollonius." Amer. Math. Monthly 75, 5-15, 1968.Coxeter, H. S. M. Introduction to Geometry, 2nd ed. New York: Wiley, pp. 13-14, 1969.Dergiades, N. "The Soddy Circles." Forum Geometricorum 7, 191-197, 2007. http://forumgeom.fau.edu/FG2007volume7/FG200726index.html.Dodds, P. S. and Weitz, J. S. "Packing-Limited Growth." Phys. Rev. E 65, 056108, 2002.Elkies, N. D. and Fukuta, J. "Problem E3236 and Solution." Amer. Math. Monthly 97, 529-531, 1990.Eppstein, D. "Tangent Spheres and Triangle Centers." Amer. Math. Monthly 108, 63-66, 2001.Fillmore, J. P. and Paluszny, M. Seminarberichte Mathematik Fernuniversität Hagen 62, 45, 1997.Gardner, M. Fractal Music, Hypercards, and More: Mathematical Recreations from Scientific American Magazine. New York: W. H. Freeman, 1992.Gosper, R. W. "Soddy's Theorem on Mutually Tangent Circles, Generalized to n Dimensions." http://www.ippi.com/rwg/Sodddy.htm.Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H. "Apollonian Circle Packings: Geometry and Group Theory I. The Apollonian Group." 15 Jul 2004. http://arxiv.org/abs/math/0010298.Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H. "Apollonian Circle Packings: Geometry and Group Theory II. Super-Apollonian Group and Integral Packings." 15 Jul 2004. http://arxiv.org/abs/math/0010302.Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H. "Apollonian Circle Packings: Geometry and Group Theory III. Higher Dimensions." 16 Jun 2004. http://arxiv.org/abs/math/0010324.Graham, R. L.; Lagarias, J. C.; Mallows, C. L.; Wilks, A. R.; and Yan, C. H. "Apollonian Circle Packings: Number Theory." J. Number Th. 100, 1-45, 2003.Kimberling, C. "Central Points and Central Lines in the Plane of a Triangle." Math. Mag. 67, p. 181, 1994.Kimberling, C. "Triangle Centers and Central Triangles." Congr. Numer. 129, 1-295, 1998."The Kiss Precise." Nature 139, 62, 1937.Martini, H. In Geometrie und ihre Anwendungen in Kunst, Natur und Technik (Ed. O. Giering and J. Hoschek). Munich, Germany: Carl Hanser, 1994.Oldknow, A. "Computer Aided Research into the Geometry of the Triangle." Math. Gaz. 79, 263-274, 1995.Söderberg, B. Phys. Rev. A 46, 1859, 1992.Soddy, F. "The Kiss Precise." Nature 137, 1021, 1936.Study, E. Math. Ann. 49, 497, 1897.Vandeghen, A. "Soddy's Circles and the De Longchamps Point of a Triangle." Amer. Math. Monthly 71, 176-179, 1964.Veldkamp, G. R. "A Theorem Concerning Soddy-Circles." Elem. Math. 21, 15-17, 1966.Veldkamp, G. R. "The Isoperimetric Point and the Point(s) of Equal Detour." Amer. Math. Monthly 92, 546-558, 1985.Wells, D. The Penguin Dictionary of Curious and Interesting Geometry. London: Penguin, pp. 4-5, 1991.

在 中被引用

索迪圓

請這樣引用

Weisstein, Eric W. “索迪圓.” 來自 Web 資源。 https://mathworld.tw/SoddyCircles.html

主題分類