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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 Sectrix of Maclaurin ： ウィキペディア英語版 Sectrix of Maclaurin In geometry, a sectrix of Maclaurin is defined as the curve swept out by the point of intersection of two lines which are each revolving at constant rates about different points called poles. Equivalently, a sectrix of Maclaurin can be defined as a curve whose equation in biangular coordinates is linear. The name is derived from the trisectrix of Maclaurin (named for Colin Maclaurin), which is a prominent member of the family, and their sectrix property, which means they can be used to divide an angle into a given number of equal parts. There are special cases are also known as arachnida or araneidans because of their spider-like shape, and Plateau curves after Joseph Plateau who studied them. ==Equations in polar coordinates== We are given two lines rotating about two poles $P$ and $P\_1$. By translation and rotation we may assume $P\; =\; (0,0)$ and $P\_1\; =\; (a,\; 0)$. At time $t$, the line rotating about $P$ has angle $\backslash theta\; =\; \backslash kappa\; t\; +\; \backslash alpha$ and the line rotating about $P\_1$ has angle $\backslash theta\_1\; =\; \backslash kappa\_1\; t\; +\; \backslash alpha\_1$, where $\backslash kappa$, $\backslash alpha$, $\backslash kappa\_1$ and $\backslash alpha\_1$ are constants. Eliminate $t$ to get $\backslash theta\_1\; =\; q\; \backslash theta\; +\; \backslash theta\_0$ where $q\; =\; \backslash kappa\_1\; /\; \backslash kappa$ and $\backslash theta\_0\; =\; \backslash alpha\_1\; -\; q\; \backslash alpha$. We assume $q$ is rational, otherwise the curve is not algebraic and is dense in the plane. Let $Q$ be the point of intersection of the two lines and let $\backslash psi$ be the angle at $Q$, so $\backslash psi\; =\; \backslash theta\_1\; -\; \backslash theta$. If $r$ is the distance from $P$ to $Q$ then, by the law of sines, :$=\; \backslash !$ so :$r\; =\; a\; \backslash frac\; =\; a\; \backslash frac\; \backslash !$ is the equation in polar coordinates. The case $\backslash theta\_0\; =\; 0$ and $q\; =\; n$ where $n$ is an integer greater than 2 gives arachnida or araneidan curves :$r\; =\; a\; \backslash frac\; \backslash !$ The case $\backslash theta\_0\; =\; 0$ and $q\; =\; -n$ where $n$ is an integer greater than 1 gives alternate forms of arachnida or araneidan curves :$r\; =\; a\; \backslash frac\; \backslash !$ A similar derivation to that above gives :$r\_1\; =\; (-a)\; \backslash frac\; \backslash !$ as the polar equation (in $r\_1$ and $\backslash theta\_1$) if the origin is shifted to the right by $a$. Note that this is the earlier equation with a change of parameters; this to be expected from the fact that two poles are interchangeable in the construction of the curve. 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「Sectrix of Maclaurin」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.