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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Deltoid curve ： ウィキペディア英語版 Deltoid curve In geometry, a deltoid, also known as a tricuspoid or Steiner curve, is a hypocycloid of three cusps. In other words, it is the roulette created by a point on the circumference of a circle as it rolls without slipping along the inside of a circle with three or one-and-a-half times its radius. It is named after the Greek letter delta which it resembles. More broadly, a deltoid can refer to any closed figure with three vertices connected by curves that are concave to the exterior, making the interior points a non-convex set. () ==Equations== A deltoid can be represented (up to rotation and translation) by the following parametric equations :$x=2a\backslash cos(t)+a\backslash cos(2t)\; \backslash ,$ :$y=2a\backslash sin(t)-a\backslash sin(2t)\backslash ,$ where ''a'' is the radius of the rolling circle. In complex coordinates this becomes :$z=2ae^+ae^$. The variable ''t'' can be eliminated from these equations to give the Cartesian equation :$(x^2+y^2)^2+18a^2(x^2+y^2)-27a^4\; =\; 8a(x^3-3xy^2),\backslash ,$ so the deltoid is a plane algebraic curve of degree four. In polar coordinates this becomes :$r^4+18a^2r^2-27a^4=8ar^3\backslash cos\; 3\backslash theta\backslash ,.$ The curve has three singularities, cusps corresponding to $t=0,\backslash ,\; \backslash pm\backslash tfrac$. The parameterization above implies that the curve is rational which implies it has genus zero. A line segment can slide with each end on the deltoid and remain tangent to the deltoid. The point of tangency travels around the deltoid twice while each end travels around it once. The dual curve of the deltoid is :$x^3-x^2-(3x+1)y^2=0,\backslash ,$ which has a double point at the origin which can be made visible for plotting by an imaginary rotation y ↦ iy, giving the curve :$x^3-x^2+(3x+1)y^2=0\backslash ,$ with a double point at the origin of the real plane. 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「Deltoid curve」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.