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In geometry, the multifocal ellipse (also known as ''n''-ellipse, ''k''-ellipse, polyellipse, egglipse, generalized ellipse, and (in German) Tschirnhaus'sche Eikurve) is a generalization of the ellipse allowing more than two foci. Specifically, given ''n'' points (''u''''i'', ''v''''i'') in a plane (foci), an ''n''-ellipse is the locus of all points of the plane whose sum of distances to the ''n'' foci is a constant ''d''. The set of points of an ''n''-ellipse is defined as: : The 1-ellipse corresponds to the circle. The 2-ellipse corresponds to the classic ellipse. Both are algebraic curves of degree 2. For any number of foci, the curves are convex and closed. If ''n'' is odd, the algebraic degree of the curve is while if ''n'' is even the degree is 〔J. Nie, P.A. Parrilo, B. Sturmfels: "(Semidefinite representation of the ''k''-ellipse )".〕 In the ''n''=3 case, the curve is smooth unless it goes through a focus.〔 == References == *James Clerk Maxwell: "(Paper on the Description of Oval Curves ), Feb 1846, from ''The Scientific Letters and Papers of James Clerk Maxwell: 1846-1862'' *Z.A. Melzak and J.S. Forsyth: "Polyconics 1. polyellipses and optimization", ''Q. of Appl. Math.'', pages 239–255, 1977. *P.L. Rosin: "(On the Construction of Ovals )" *P.V. Sahadevan: "The theory of egglipse—a new curve with three focal points", ''International Journal of Mathematical Education in Science and Technology'' 18 (1987), 29–39. MR 88b:51041; Zbl 613.51030 *J. Sekino: "''n''-Ellipses and the Minimum Distance Sum Problem", ''American Mathematical Monthly'' 106 #3 (March 1999), 193–202. MR 2000a:52003; Zbl 986.51040. *B. Sturmfels: "(The Geometry of Semidefinite Programming )", pp. 9-16. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「N-ellipse」の詳細全文を読む スポンサード リンク
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