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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 cissoid ： ウィキペディア英語版 cissoid In geometry, a cissoid is a curve generated from two given curves ''C''1, ''C''2 and a point ''O'' (the pole). Let ''L'' be a variable line passing through ''O'' and intersecting ''C''1 at ''P''1 and ''C''2 at ''P''2. Let P be the point on L so that ''OP'' = ''P''1''P''2. (There are actually two such points but P is chosen so that ''P'' is in the same direction from ''O'' as ''P''2 is from ''P''1.) Then the locus of such points ''P'' is defined to be the cissoid of the curves ''C''1, ''C''2 relative to ''O''. Slightly different but essentially equivalent definitions are used by different authors. For example, ''P'' may be defined to be the point so that ''OP'' = ''OP''1 + ''OP''2. This is equivalent to the other definition if ''C''1 is replaced by its reflection through ''O''. Or ''P'' may be defined as the midpoint of ''P''1 and ''P''2; this produces the curve generated by the previous curve scaled by a factor of 1/2. The word "cissoid" comes from the Greek ''kissoeidēs'' "ivy shaped" from ''kissos'' "ivy" and -''oeidēs'' "having the likeness of". ==Equations== If ''C''1 and ''C''2 are given in polar coordinates by $r=f\_1(\backslash theta)$ and $r=f\_2(\backslash theta)$ respectively, then the equation $r=f\_2(\backslash theta)-f\_1(\backslash theta)$ describes the cissoid of ''C''1 and ''C''2 relative to the origin. However, because a point may be represented in multiple ways in polar coordinates, there may be other branches of the cissoid which have a different equation. Specifically, ''C''1 is also given by :$r=-f\_1(\backslash theta+\backslash pi),\backslash \; r=-f\_1(\backslash theta-\backslash pi),\backslash \; r=f\_1(\backslash theta+2\backslash pi),\backslash \; r=f\_1(\backslash theta-2\backslash pi),\backslash \; \backslash dots$. So the cissoid is actually the union of the curves given by the equations :$r=f\_2(\backslash theta)-f\_1(\backslash theta),\backslash \; r=f\_2(\backslash theta)+f\_1(\backslash theta+\backslash pi),\backslash \; r=f\_2(\backslash theta)+f\_1(\backslash theta-\backslash pi),\backslash$ :$r=f\_2(\backslash theta)-f\_1(\backslash theta+2\backslash pi),\backslash \; r=f\_2(\backslash theta)-f\_1(\backslash theta-2\backslash pi),\backslash \; \backslash dots$. It can be determined on an individual basis depending on the periods of ''f''1 and ''f''2, which of these equations can be eliminated due to duplication. For example, let ''C''1 and ''C''2 both be the ellipse :$r=\backslash frac$. The first branch of the cissoid is given by :$r=\backslash frac-\backslash frac=0$, which is simply the origin. The ellipse is also given by :$r=\backslash frac$, so a second branch of the cissoid is given by :$r=\backslash frac+\backslash frac$ which is an oval shaped curve. If each ''C''1 and ''C''2 are given by the parametric equations :$x\; =\; f\_1(p),\backslash \; y\; =\; px$ and :$x\; =\; f\_2(p),\backslash \; y\; =\; px$, then the cissoid relative to the origin is given by :$x\; =\; f\_2(p)-f\_1(p),\backslash \; y\; =\; px$. 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「cissoid」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.