曲面 (きょくめん)とは【ピクシブ百科事典】

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src="https://pixon.ads-pixiv.net/show?zone_id=d_header&format=js&s=0&up=0&ng=w&os=and&ref=dic.pixiv.net%2Fa%2F%25E6%259B%25B2%25E9%259D%25A2&ab_test_digits_first=&num=fdadbf1d"></script></section> <article id="content"> <header> <div id="main-image" class="embedimage"> <a href="https://www.pixiv.net/artworks/3553887" title="曲面（練習）" target="_blank" data-author-name="chibaf" data-author-url="https://www.pixiv.net/artworks/3553887" class="illust" gtm-class="main-illust" gtm-id="3553887" > <img src="https://i.pximg.net/c/600x600/img-master/img/2009/03/22/14/31/57/3553887_p0_master1200.jpg" alt="曲面（練習）"/> </a> </div> <div id="content_title" class="info"> <h1 id="article-name">曲面</h1> <p class="subscript">きょくめん</p> <div class="summary"> 曲がっている面。曲線の面バージョン。 </div> <ul id="share-buttons" class="share-buttons"> <li class="twitter"></li> </ul> <div class="link-to-pixiv"> <p> <a href="https://www.pixiv.net/tags/%E6%9B%B2%E9%9D%A2/illustrations" target="_blank" gtm-class="article-top_view-illust-tag" gtm-id="曲面" >pixivで「曲面」のイラストを見る</a> </p> <p> <a href="https://www.pixiv.net/tags/%E6%9B%B2%E9%9D%A2/novels" gtm-class="article-top_view-novel-tag" gtm-id="曲面" target="_blank" >pixivで「曲面」の小説を読む</a> </p> <div class="gtm-illust-create-in-pixpedia-article"> <p> <a href="https://www.pixiv.net/upload.php?tag=%E6%9B%B2%E9%9D%A2" target="_blank" rel="nofollow noopener" gtm-class="article-top_create-illust" gtm-id="曲面" >pixivで「曲面」のイラストを投稿する</a> </p> <p> <a href="https://www.pixiv.net/novel/upload.php?tag=%E6%9B%B2%E9%9D%A2" target="_blank" rel="nofollow noopener" gtm-class="article-top_create-novel" gtm-id="曲面" >pixivで「曲面」の小説を投稿する</a> </p> </div> </div> </div> </header> <div id="main"> <section id="article-body"> <div class='article_index'> <div class='index_title'><span>目次</span> [<a href='#' class='toggle_index'>非表示</a>] </div> <div class='index_body'> <li class="index_level1"> <a href="#h2_0" gtm-class="article-toc_click" gtm-id="概要"> 1 概要 </a> </li> <li class="index_level1"> <a href="#h2_1" gtm-class="article-toc_click" gtm-id="滑らかな曲面"> 2 滑らかな曲面 </a> </li> <li class="index_level1"> <a href="#h2_2" gtm-class="article-toc_click" gtm-id="種類"> 3 種類 </a> </li> <li class="index_level1"> <a href="#h2_3" gtm-class="article-toc_click" gtm-id="関連タグ"> 4 関連タグ </a> </li> <li class="index_level2"> <a href="#h3_0" gtm-class="article-toc_click" gtm-id="曲面を持つ立体"> 4.1 曲面を持つ立体 </a> </li> <li class="index_level1"> <a href="#h2_4" gtm-class="article-toc_click" gtm-id="関連リンク"> 5 関連リンク </a> </li> </div> </div> <div style="clear:both;"></div> <h2 id="h2_0"><span>概要</span><a href="/ae/%E6%9B%B2%E9%9D%A2?heading=h2&id=0"data-first-header="true"data-tooltip="この見出しを編集"class="link_edit_partial"data-gtm-class="article-heading_edit"data-gtm-id="h2_0"data-google-interstitial="false"rel="nofollow"><img src="https://s.pximg.net/source/dic/images/icon_edit_partial.svg?20120424" alt="編集" width="24" height="24"></a></h2><p>曲がっている<a href="/a/%E9%9D%A2" gtm-class="article-body_article" gtm-id="面">面</a>であり、<a href="/a/%E6%9B%B2%E7%B7%9A" gtm-class="article-body_article" gtm-id="曲線">曲線</a>の面バージョンである。</p><p>ここで言う「曲がっている」というのは、<a href="/a/%E9%81%93%E8%B7%AF" gtm-class="article-body_article" gtm-id="道路">道路</a>が曲がってる場合のような平坦な曲がり方ではなく、<a href="/a/%E5%B1%B1" gtm-class="article-body_article" gtm-id="山">山</a>の起伏のような<a href="/a/%E4%B8%89%E6%AC%A1%E5%85%83" gtm-class="article-body_article" gtm-id="三次元">三次元</a>的な曲がり方を指す。</p><p>そういう曲がり方は三次元以上の<a href="/a/%E7%A9%BA%E9%96%93" gtm-class="article-body_article" gtm-id="空間">空間</a>でなければ起こらないので、三次元以上でのみ存在する。</p><p><a href="/a/%E9%9D%A2%E7%A9%8D" gtm-class="article-body_article" gtm-id="面積">面積</a>で表される<a href="/a/%E4%BA%8C%E6%AC%A1%E5%85%83" gtm-class="article-body_article" gtm-id="二次元">二次元</a>的な存在であり、通常は<a href="/a/%E7%AB%8B%E4%BD%93" gtm-class="article-body_article" gtm-id="立体">立体</a>とは見なされないが、<a href="/a/%E7%90%83" gtm-class="article-body_article" gtm-id="球">球</a>や<a href="/a/%E3%83%88%E3%83%BC%E3%83%A9%E3%82%B9" gtm-class="article-body_article" gtm-id="トーラス">トーラス</a>のように、曲面なのか立体なのか曖昧な使われ方をするものも多い。</p><br /><p>曲がっていなければ「<a href="/a/%E5%B9%B3%E9%9D%A2" gtm-class="article-body_article" gtm-id="平面">平面</a>」となるが、曲線の場合と同様、文脈によっては「平面とは限らない面」という意味合いで、平面や折れ面も曲面の一種となる。</p><p>平面を含む滑らかな曲面は特に「可微分曲面」と呼ばれ、更に折れてる面も含んだようなものは「位相曲面」と呼ばれる。</p><br /><h2 id="h2_1"><span>滑らかな曲面</span><a href="/ae/%E6%9B%B2%E9%9D%A2?heading=h2&id=1"data-tooltip="この見出しを編集"class="link_edit_partial"data-gtm-class="article-heading_edit"data-gtm-id="h2_1"data-google-interstitial="false"rel="nofollow"><img src="https://s.pximg.net/source/dic/images/icon_edit_partial.svg?20120424" alt="編集" width="24" height="24"></a></h2><p>曲面の曲がり方は、曲線と同様の<a class="no-article" href="/a/%E6%9B%B2%E7%8E%87">曲率</a>を利用して表す事ができる。</p><p>曲線の場合と同様、滑らかな曲面の一点には接面を定義できる。</p><p>ただ、曲面の場合は、同じ地点でも向きによって曲がり方が異なっている事が殆どである。</p><br /><p>また、<a href="/a/%E5%86%86%E6%9F%B1" gtm-class="article-body_article" gtm-id="円柱">円柱</a>や<a href="/a/%E5%86%86%E9%8C%90" gtm-class="article-body_article" gtm-id="円錐">円錐</a>における曲面のように<a href="/a/%E5%B1%95%E9%96%8B%E5%9B%B3" gtm-class="article-body_article" gtm-id="展開図">展開図</a>を作る事が可能なタイプと、球における曲面のように展開図を作れないタイプとがある。</p><p>この辺りの性質を記述するものとして、「ガウス曲率」というものがある。</p><p>これは方向による曲率の最大と最小を<a href="/a/%E6%8E%9B%E3%81%91%E7%AE%97" gtm-class="article-body_article" gtm-id="掛け算">掛け算</a>したもののようになっている。</p><p>円柱や円錐の面はこれが0であり、それが展開図を作れる条件にもなっている。</p><p>一方、球の面はこれが正となっており、負ならば<a href="/a/%E9%9E%8D" gtm-class="article-body_article" gtm-id="鞍">鞍</a>の中心部のような曲がり方となる。</p><br /><h2 id="h2_2"><span>種類</span><a href="/ae/%E6%9B%B2%E9%9D%A2?heading=h2&id=2"data-tooltip="この見出しを編集"class="link_edit_partial"data-gtm-class="article-heading_edit"data-gtm-id="h2_2"data-google-interstitial="false"rel="nofollow"><img src="https://s.pximg.net/source/dic/images/icon_edit_partial.svg?20120424" alt="編集" width="24" height="24"></a></h2><ul><li><a href="/a/%E7%90%83%E9%9D%A2" gtm-class="article-body_article" gtm-id="球面">球面</a>…その名の通り、<a href="/a/%E7%90%83" gtm-class="article-body_article" gtm-id="球">球</a>の<a href="/a/%E8%A1%A8%E9%9D%A2" gtm-class="article-body_article" gtm-id="表面">表面</a>。</li><li><a class="no-article" href="/a/%E6%94%BE%E7%89%A9%E9%9D%A2">放物面</a>…<a href="/a/%E6%94%BE%E7%89%A9%E7%B7%9A" gtm-class="article-body_article" gtm-id="放物線">放物線</a>を回転させる事でできる面。<a href="/a/%E3%83%91%E3%83%A9%E3%83%9C%E3%83%A9%E3%82%A2%E3%83%B3%E3%83%86%E3%83%8A" gtm-class="article-body_article" gtm-id="パラボラアンテナ">パラボラアンテナ</a>の語源。</li><li><a class="no-article" href="/a/%E5%8F%8C%E6%9B%B2%E9%9D%A2">双曲面</a>…<a href="/a/%E5%8F%8C%E6%9B%B2%E7%B7%9A" gtm-class="article-body_article" gtm-id="双曲線">双曲線</a>を回転させる事でできる面。</li><li><a class="no-article" href="/a/%E6%A5%95%E5%86%86%E9%9D%A2">楕円面</a>…<a href="/a/%E6%A5%95%E5%86%86%E4%BD%93" gtm-class="article-body_article" gtm-id="楕円体">楕円体</a>の表面であり、<a href="/a/%E6%A5%95%E5%86%86" gtm-class="article-body_article" gtm-id="楕円">楕円</a>の形をした面ではない。</li><li>トーラス面…<a href="/a/%E3%83%88%E3%83%BC%E3%83%A9%E3%82%B9" gtm-class="article-body_article" gtm-id="トーラス">トーラス</a>の面だが、単に「トーラス」と言った場合にはこちらを指す事も多いらしい。対して、中身の詰まったものは「トーラス体」と呼ばれる。</li><li><a href="/a/%E3%83%A1%E3%83%93%E3%82%A6%E3%82%B9%E3%81%AE%E8%BC%AA" gtm-class="article-body_article" gtm-id="メビウスの輪">メビウスの輪</a>…表と裏の区別が無い帯。</li><li><a href="/a/%E3%82%AF%E3%83%A9%E3%82%A4%E3%83%B3%E3%81%AE%E5%A3%BA" gtm-class="article-body_article" gtm-id="クラインの壺">クラインの壺</a>…メビウスの輪の発展型的なもの。<a href="/a/%E5%9B%9B%E6%AC%A1%E5%85%83" gtm-class="article-body_article" gtm-id="四次元">四次元</a>的な構造を持っている。</li></ul><br /><h2 id="h2_3"><span>関連タグ</span><a href="/ae/%E6%9B%B2%E9%9D%A2?heading=h2&id=3"data-tooltip="この見出しを編集"class="link_edit_partial"data-gtm-class="article-heading_edit"data-gtm-id="h2_3"data-google-interstitial="false"rel="nofollow"><img src="https://s.pximg.net/source/dic/images/icon_edit_partial.svg?20120424" alt="編集" width="24" height="24"></a></h2><p><a href="/a/%E9%9D%A2" gtm-class="article-body_article" gtm-id="面">面</a>　<a href="/a/%E5%B9%B3%E9%9D%A2" gtm-class="article-body_article" gtm-id="平面">平面</a>　<a href="/a/%E5%81%B4%E9%9D%A2" gtm-class="article-body_article" gtm-id="側面">側面</a>　<a href="/a/%E8%A1%A8%E9%9D%A2" gtm-class="article-body_article" gtm-id="表面">表面</a>　<a href="/a/%E6%9B%B2%E3%81%8C%E3%82%8B" gtm-class="article-body_article" gtm-id="曲がる">曲がる</a>　<a class="no-article" href="/a/%E6%9B%B2%E7%8E%87">曲率</a>　<a href="/a/%E6%9B%B2%E7%B7%9A" gtm-class="article-body_article" gtm-id="曲線">曲線</a></p><p><a href="/a/%E7%AB%8B%E4%BD%93" gtm-class="article-body_article" gtm-id="立体">立体</a>　<a href="/a/%E9%87%8D%E5%8A%9B" gtm-class="article-body_article" gtm-id="重力">重力</a>　<a href="/a/%E6%9B%B2%E7%B7%9A%E7%BE%8E" gtm-class="article-body_article" gtm-id="曲線美">曲線美</a>　<a href="/a/%E6%9B%B2%E8%80%85" gtm-class="article-body_article" gtm-id="曲者">曲者</a>　<a href="/a/%E9%96%A2%E6%95%B0" gtm-class="article-body_article" gtm-id="関数">関数</a></p><p><a href="/a/%E4%BA%8C%E6%AC%A1%E5%85%83" gtm-class="article-body_article" gtm-id="二次元">二次元</a>　<a href="/a/%E4%B8%89%E6%AC%A1%E5%85%83" gtm-class="article-body_article" gtm-id="三次元">三次元</a>　<a href="/a/%E5%9B%9B%E6%AC%A1%E5%85%83" gtm-class="article-body_article" gtm-id="四次元">四次元</a>　<a href="/a/%E5%9B%B3%E5%BD%A2" gtm-class="article-body_article" gtm-id="図形">図形</a>　<a href="/a/%E5%B9%BE%E4%BD%95%E5%AD%A6" gtm-class="article-body_article" gtm-id="幾何学">幾何学</a>　<a href="/a/%E6%95%B0%E5%AD%A6" gtm-class="article-body_article" gtm-id="数学">数学</a></p><br /><h3 id="h3_0"><span>曲面を持つ立体</span><a href="/ae/%E6%9B%B2%E9%9D%A2?heading=h3&id=0"data-tooltip="この見出しを編集"class="link_edit_partial"data-gtm-class="article-heading_edit"data-gtm-id="h3_0"data-google-interstitial="false"rel="nofollow"><img src="https://s.pximg.net/source/dic/images/icon_edit_partial.svg?20120424" alt="編集" width="24" height="24"></a></h3><p><a href="/a/%E7%90%83" gtm-class="article-body_article" gtm-id="球">球</a>　<a href="/a/%E5%86%86%E9%8C%90" gtm-class="article-body_article" gtm-id="円錐">円錐</a>　<a href="/a/%E5%86%86%E6%9F%B1" gtm-class="article-body_article" gtm-id="円柱">円柱</a>　<a href="/a/%E5%8F%8C%E5%86%86%E9%8C%90" gtm-class="article-body_article" gtm-id="双円錐">双円錐</a>　<a href="/a/%E8%A7%92%E7%90%83" gtm-class="article-body_article" gtm-id="角球">角球</a>　<a href="/a/%E6%A5%95%E5%86%86%E4%BD%93" gtm-class="article-body_article" gtm-id="楕円体">楕円体</a>　<a class="no-article" href="/a/%E8%B6%852%E6%AC%A1%E6%A5%95%E5%86%86%E4%BD%93">超2次楕円体</a>/<a class="no-article" href="/a/%E8%B6%85%E4%BA%8C%E6%AC%A1%E6%A5%95%E5%86%86%E4%BD%93">超二次楕円体</a>　<a href="/a/%E3%83%88%E3%83%BC%E3%83%A9%E3%82%B9" gtm-class="article-body_article" gtm-id="トーラス">トーラス</a></p><br /><h2 id="h2_4"><span>関連リンク</span><a href="/ae/%E6%9B%B2%E9%9D%A2?heading=h2&id=4"data-tooltip="この見出しを編集"class="link_edit_partial"data-gtm-class="article-heading_edit"data-gtm-id="h2_4"data-google-interstitial="false"rel="nofollow"><img src="https://s.pximg.net/source/dic/images/icon_edit_partial.svg?20120424" alt="編集" width="24" height="24"></a></h2><ul><li><a class="external-link" target="_blank" rel="noopener nofollow ugc" href="https://ja.wikipedia.org/wiki/%E6%9B%B2%E9%9D%A2" gtm-class="article-body_external">曲面 - Wikipedia</a></li></ul> </section> <section id="footer_inline_ad" style=" display: flex; flex-direction: column; align-items: center; padding: 16px 7px; background: #FAFAFA; width: 640px; ">  <div id='footer_inline_ad_inner' style=' min-width: 300px; '> <script id="footer_inline_ad_script" type="text/javascript" src="https://pixon.ads-pixiv.net/show?zone_id=d_footer_inline&format=js&s=0&up=0&ng=w&os=and&ref=dic.pixiv.net%2Fa%2F%25E6%259B%25B2%25E9%259D%25A2&ab_test_digits_first=&num=76eced73"></script> <script id="footer_inline_ad_script_cmd" type="text/javascript"> // 配信された広告が枠をはみ出す場合は縮小する const adFrame = document.getElementById('footer_inline_ad_inner'); const resizeObserver = new ResizeObserver(() => 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