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https:///mymemo.xyz/wiki/api.php?action=feedcontributions&user=23.92.221.214&feedformat=atom 2024-05-07T05:04:21Z 利用者の投稿記録 MediaWiki 1.31.0 2016-07-13T16:53:03Z

<p>23.92.221.214: </p> <hr /> <div>{{出典の明記|date=2015年5月}}<br /> [[File:Triaxial Ellipsoid.jpg|thumb|楕円体（c&gt;b&gt;aの場合）]]<br /> [[Image:Gnuplot ellipsoid.svg|thumb|[[gnuplot]]による楕円体の描画例]]<br /> &#039;&#039;&#039;楕円体&#039;&#039;&#039;（だえんたい、ellipsoid）とは[[楕円]]を三次元へ拡張したような[[図形]]であり、その表面は[[二次曲面]]である。楕円面の[[方程式]]は<br /> :&lt;math&gt;\frac{x^2}{a^2} + \frac{y^2}{b^2} + \frac{z^2}{c^2} = 1&lt;/math&gt;<br /> である。ここで &#039;&#039;a&#039;&#039;, &#039;&#039;b&#039;&#039;, &#039;&#039;c&#039;&#039; はそれぞれx軸、y軸、z軸方向の[[径]]の半分の長さに相当する。なお &#039;&#039;a&#039;&#039; = &#039;&#039;b&#039;&#039; = &#039;&#039;c&#039;&#039; である楕円体は[[球]]である。また &#039;&#039;a&#039;&#039;, &#039;&#039;b&#039;&#039;, &#039;&#039;c&#039;&#039; のうちいずれか2つが等しい楕円体は楕円の軸を中心に楕円を回転して得られる[[回転体]]であり、[[回転楕円体]]と呼ばれる。楕円体は球と同様にxy平面、yz平面、zx平面に関して[[対称]]である。<br /> <br /> == 楕円体の性質を表す式 ==<br /> 楕円面の[[媒介変数表示]]は[[極座標系]]を用いると<br /> :&lt;math&gt;\begin{align}x&amp;=a \sin \theta \cos \varphi\\<br /> y&amp;=b \sin \theta \sin \varphi\\<br /> z&amp;=c \cos \theta \end{align}&lt;/math&gt;<br /> <br /> :&lt;math&gt;0 \le \theta \le \pi ,\quad 0 \le \varphi \le 2 \pi&lt;/math&gt;<br /> と表される。楕円体の[[体積]] &#039;&#039;V&#039;&#039; は<br /> :&lt;math&gt;V = \frac{4}{3} \pi abc&lt;/math&gt;<br /> である。[[表面積]] &#039;&#039;S&#039;&#039; は<br /> :&lt;math&gt;S = 2\pi\left(c^2+b\sqrt{a^2-c^2}E(o\!\varepsilon,m)+\frac{bc^2}{\sqrt{a^2-c^2}}F(o\!\varepsilon,m)\right)&lt;/math&gt;<br /> となる。&lt;math&gt;o\!\varepsilon&lt;/math&gt;はモジュラー角、&lt;math&gt;m=\frac{b^2-c^2}{b^2 \sin^2 o\!\varepsilon}&lt;/math&gt;、&lt;math&gt;E(o\!\varepsilon,m)&lt;/math&gt;、&lt;math&gt;F(o\!\varepsilon,m)&lt;/math&gt; はそれぞれ第一種および第二種[[楕円積分]]である。近似式で<br /> :&lt;math&gt;S \approx 4\pi\!\left(\frac{ a^p b^p+a^p c^p+b^p c^p }{3}\right)^{1/p}&lt;/math&gt;<br /> という公式が知られている。ここでpは定数で、&#039;&#039;p&#039;&#039; = 1.6075 のとき[[誤差]]は最大でも1.061%である。<br /> <br /> == 関連項目 ==<br /> {{commonscat|Ellipsoids}}<br /> * [[楕円]]<br /> * [[楕円積分]]<br /> * [[双曲面]]<br /> <br /> {{DEFAULTSORT:たえんたい}}<br /> [[Category:曲面]]<br /> [[Category:数学に関する記事]]</div>

23.92.221.214
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