{"id":501,"date":"2012-12-03T14:15:00","date_gmt":"2012-12-03T05:15:00","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=501"},"modified":"2021-09-27T21:10:58","modified_gmt":"2021-09-27T12:10:58","slug":"thr201006","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/thr201006\/","title":{"rendered":"\u6771\u5317\u5927\u7406\u7cfb2010\uff1a\u7b2c6\u554f"},"content":{"rendered":"
\n

\\(xy\\) \u5e73\u9762\u306b\u304a\u3044\u3066, \u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3057 P \\((1,0)\\) \u3092\u9802\u70b9\u306e \\(1\\) \u3064\u3068\u3059\u308b\u6b63 \\(6\\) \u89d2\u5f62\u3092 \\(X\\) \u3068\u3059\u308b.\r\n\\(A\\) \u3092 \\(2\\) \u6b21\u306e\u6b63\u65b9\u884c\u5217\u3068\u3057, \\(X\\) \u306e\u5404\u9802\u70b9 \\((x,y)\\) \u306b\u5bfe\u3057\u3066, \u884c\u5217 \\(A\\) \u306e\u8868\u3059\u79fb\u52d5\r\n\\[\r\n\\left( \\begin{array}{c} x' \\\\ y' \\end{array} \\right) = A \\left( \\begin{array}{c} x \\\\ y \\end{array} \\right)\r\n\\]\r\n\u3067\u5f97\u3089\u308c\u308b\u70b9 \\((x' , y')\\) \u306f \\(X\\) \u306e\u8fba\u4e0a\u306e\u70b9\uff08\u9802\u70b9\u3092\u542b\u3080\uff09\u3067\u3042\u308b\u3068\u3059\u308b.\r\n\u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u70b9 P \u304c\u884c\u5217 \\(A\\) \u306e\u8868\u3059\u79fb\u52d5\u3067 P \u81ea\u8eab\u306b\u79fb\u308b\u3068\u304d, \\(X\\) \u306e\u5404\u9802\u70b9\u306f \\(X\\) \u306e\u3044\u305a\u308c\u304b\u306e\u9802\u70b9\u306b\u79fb\u308b\u3053\u3068\u3092\u793a\u305b. \u307e\u305f, \u305d\u306e\u3068\u304d\u306e\u884c\u5217 \\(A\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u70b9 P \u304c\u884c\u5217 \\(A\\) \u306e\u8868\u3059\u79fb\u52d5\u3067 \\(X\\) \u306e\u3042\u308b\u9802\u70b9\u306b\u79fb\u308b\u3068\u304d, \\(X\\) \u306e\u5404\u9802\u70b9\u306f \\(X\\) \u306e\u3044\u305a\u308c\u304b\u306e\u9802\u70b9\u306b\u79fb\u308b\u3053\u3068\u3092\u793a\u305b. \u307e\u305f, \u305d\u306e\u3068\u304d\u306e\u884c\u5217 \\(A\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h2>\r\n

    (1)<\/strong><\/p>\r\n

    \\(X\\) \u306e\u5404\u9802\u70b9\u3092 \\(\\text{P} {} _ k \\, \\left( \\cos \\dfrac{k \\pi}{3} , \\sin \\dfrac{k \\pi}{3} \\right) \\ ( k = 0, 1, \\cdots , 5 )\\) \u3068\u304a\u304f.<\/p>\r\n\"\"\r\n

    \u70b9 \\(\\text{P} {} _ k\\) \u304c \\(A\\) \u306e\u79fb\u52d5\u306b\u3088\u3063\u3066\u79fb\u308b\u70b9\u3092 \\(\\text{P'} {} _ k\\) \u3068\u304a\u304f.
    \r\n\\(X\\) \u306f\u6b63 \\(6\\) \u89d2\u5f62\u306a\u306e\u3067\r\n\\[\r\n\\overrightarrow{\\text{P} {} _ 2\\text{P} {} _ 1} = \\overrightarrow{\\text{OP}} \\quad ... [1]\r\n\\]\r\n\u6761\u4ef6\u3088\u308a, \\(\\overrightarrow{\\text{OP}} = A \\, \\overrightarrow{\\text{OP}}\\) \u306a\u306e\u3067\r\n\\[\r\n\\overrightarrow{\\text{P'} {} _ 2\\text{P'} {} _ 1} = \\overrightarrow{\\text{OP}}\r\n\\]\r\n\u70b9 \\(\\text{P'} {} _ , \\text{P'} {} _ 2\\) \u306f \\(X\\) \u4e0a\u306e\u70b9\u3060\u304b\u3089\r\n\\[\r\n\\overrightarrow{\\text{P'} {} _ 2\\text{P'} {} _ 1} = \\overrightarrow{\\text{P} {} _ 2\\text{P} {} _ 1} , \\ \\overrightarrow{\\text{P} {} _ 4\\text{P} {} _ 5}\r\n\\]\r\n\u3088\u3063\u3066, \u70b9 \\(\\text{P} {} _ 1\\) \u306f\u70b9 \\(\\text{P} {} _ 1\\) \u307e\u305f\u306f \\(\\text{P} {} _ 5\\) \u306b\u79fb\u308b.
    \r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nA \\left( \\begin{array}{cc} 1 & \\frac{1}{2} \\\\ 0 & \\frac{\\sqrt{3}}{2} \\end{array} \\right) = \\left( \\begin{array}{cc} 1 & \\frac{1}{2} \\\\ 0 & \\pm \\frac{\\sqrt{3}}{2} \\end{array} \\right)\r\n\\]\r\n\u3053\u3053\u3067\r\n\\[\\begin{align}\r\n\\left( \\begin{array}{cc} 1 & \\frac{1}{2} \\\\ 0 & \\frac{\\sqrt{3}}{2} \\end{array} \\right)^{-1} & = \\dfrac{2}{\\sqrt{3}} \\left( \\begin{array}{cc} \\frac{\\sqrt{3}}{2} & -\\frac{1}{2} \\\\ 0 & 1 \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} 1 & -\\frac{1}{\\sqrt{3}} \\\\ 0 & \\frac{2}{\\sqrt{3}} \\end{array} \\right)\r\n\\end{align}\\]\r\n\u306a\u306e\u3067\r\n\\[\\begin{align}\r\nA & = \\left( \\begin{array}{cc} 1 & \\frac{1}{2} \\\\ 0 & \\pm \\frac{\\sqrt{3}}{2} \\end{array} \\right) \\left( \\begin{array}{cc} 1 & -\\frac{1}{\\sqrt{3}} \\\\ 0 & \\frac{2}{\\sqrt{3}} \\end{array} \\right) \\\\\r\n& = \\underline{\\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & \\pm 1 \\end{array} \\right)}\r\n\\end{align}\\]\r\n\u307e\u305f, \\(2\\) \u3064\u306e\u884c\u5217\u306f, \u79fb\u52d5\u306a\u3057\uff08\u540c\u5024\u5909\u63db\uff09\u3068 \\(x\\) \u8ef8\u5bfe\u79f0\u79fb\u52d5\u306a\u306e\u3067, \u5404\u9802\u70b9\u306f\u3044\u305a\u308c\u304b\u306e\u9802\u70b9\u306b\u79fb\u52d5\u3059\u308b.<\/p>\r\n

    (2)<\/strong><\/p>\r\n

    (1)<\/strong> \u3068\u540c\u69d8\u306b\u3057\u3066\u8003\u3048\u308b.
    \r\n\u305f\u3060\u3057, \\(\\text{P} {} _ k\\) \u3068 \\(\\text{P} {} _ {k \\pm 6}\\) \u306f\u540c\u3058\u70b9\u3067\u3042\u308b\u3068\u3059\u308b.
    \r\n\u70b9 P \u304c\u70b9 \\(\\text{P} {} _ k\\) \u306b\u79fb\u52d5\u3059\u308b\u3068\u304d,\r\n\\[\r\n\\overrightarrow{\\text{P'} {} _ 2\\text{P'} {} _ 1} = \\overrightarrow{\\text{OP} {} _ k}\r\n\\]\r\n\u70b9 \\(\\text{P'} {} _ 1\\) , \\(\\text{P'} {} _ 2\\) \u306f \\(X\\) \u4e0a\u306e\u70b9\u3060\u304b\u3089\r\n\\[\r\n\\overrightarrow{\\text{P'} {} _ 2\\text{P'} {} _ 1} = \\overrightarrow{\\text{P} {} _ {k+2}\\text{P} {} _ {k+1}} , \\ \\overrightarrow{\\text{P} {} _ {k-2}\\text{P} {} _ {k-1}}\r\n\\]\r\n\u3088\u3063\u3066, \u70b9 \\(\\text{P} {} _ 1\\) \u306f\u70b9 \\(\\text{P} {} _ {k+1}\\) \u307e\u305f\u306f \\(\\text{P} {} _ {k-1}\\) \u306b\u79fb\u308b.
    \r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\nA \\left( \\begin{array}{cc} 1 & \\frac{1}{2} \\\\ 0 & \\frac{\\sqrt{3}}{2} \\end{array} \\right) = \\left( \\begin{array}{cc} \\cos \\frac{k \\pi}{3} & \\cos \\frac{(k \\pm 1) \\pi}{3} \\\\ \\sin \\frac{k \\pi}{3} & \\sin \\frac{(k \\pm 1) \\pi}{3} \\end{array} \\right)\r\n\\]\r\n\u3086\u3048\u306b\r\n\\[\\begin{align}\r\nA & = \\left( \\begin{array}{cc} \\cos \\frac{k \\pi}{3} & \\cos \\frac{(k \\pm 1) \\pi}{3} \\\\ \\sin \\frac{k \\pi}{3} & \\sin \\frac{(k \\pm 1) \\pi}{3} \\end{array} \\right) \\left( \\begin{array}{cc} 1 & -\\frac{1}{\\sqrt{3}} \\\\ 0 & \\frac{2}{\\sqrt{3}} \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} \\cos \\frac{k \\pi}{3} & \\frac{1}{\\sqrt{3}} \\left\\{ -\\cos \\frac{k \\pi}{3} +\\cos \\frac{(k \\pm 1) \\pi}{3} \\right\\} \\\\ \\sin \\frac{k \\pi}{3} & \\frac{1}{\\sqrt{3}} \\left\\{ -\\sin \\frac{k \\pi}{3} +\\sin \\frac{(k \\pm 1) \\pi}{3} \\right\\} \\end{array} \\right)\r\n\\end{align}\\]\r\n

      \r\n

    1. 1*<\/strong>\u3000\\(k=0\\) \u306e\u3068\u304d
      \r\n(1)<\/strong> \u3067\u6c42\u3081\u305f\u901a\u308a\u3067\r\n\\[\r\nA = \\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & \\pm 1 \\end{array} \\right) = \\left( \\begin{array}{cc} \\cos 0 & \\mp \\sin 0 \\\\ \\sin 0 & \\pm \\cos 0 \\end{array} \\right)\r\n\\]<\/li>\r\n

    2. 2*<\/strong>\u3000\\(k=1\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nA & = \\left( \\begin{array}{cc} \\cos \\frac{\\pi}{3} & \\frac{1}{\\sqrt{3}} \\left\\{ -\\cos \\frac{\\pi}{3} +\\cos \\frac{(1 \\pm 1) \\pi}{3} \\right\\} \\\\ \\sin \\frac{\\pi}{3} & \\frac{1}{\\sqrt{3}} \\left\\{ -\\sin \\frac{\\pi}{3} +\\sin \\frac{(1 \\pm 1) \\pi}{3} \\right\\} \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} \\frac{1}{2} & \\mp \\frac{\\sqrt{3}}{2} \\\\ \\frac{\\sqrt{3}}{2} & \\pm \\frac{1}{2} \\end{array} \\right) = \\left( \\begin{array}{cc} \\cos \\frac{\\pi}{3} & \\mp \\sin \\frac{\\pi}{3} \\\\ \\sin \\frac{\\pi}{3} & \\pm \\cos \\frac{\\pi}{3} \\end{array} \\right)\r\n\\end{align}\\]<\/li>\r\n

    3. 3*<\/strong>\u3000\\(k=2\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nA & = \\left( \\begin{array}{cc} \\cos \\frac{2 \\pi}{3} & \\frac{1}{\\sqrt{3}} \\left\\{ -\\cos \\frac{2 \\pi}{3} +\\cos \\frac{(2 \\pm 1) \\pi}{3} \\right\\} \\\\ \\sin \\frac{2 \\pi}{3} & \\frac{1}{\\sqrt{3}} \\left\\{ -\\sin \\frac{2 \\pi}{3} +\\sin \\frac{(2 \\pm 1) \\pi}{3} \\right\\} \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} -\\frac{1}{2} & \\pm \\frac{\\sqrt{3}}{2} \\\\ \\frac{\\sqrt{3}}{2} & \\mp \\frac{1}{2} \\end{array} \\right) = \\left( \\begin{array}{cc} \\cos \\frac{2 \\pi}{3} & \\mp \\sin \\frac{2 \\pi}{3} \\\\ \\sin \\frac{2 \\pi}{3} & \\pm \\cos \\frac{2 \\pi}{3} \\end{array} \\right)\r\n\\end{align}\\]<\/li>\r\n

    4. 4*<\/strong>\u3000\\(k=3\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nA & = \\left( \\begin{array}{cc} \\cos \\pi & \\frac{1}{\\sqrt{3}} \\left\\{ -\\cos \\pi +\\cos \\frac{(3 \\pm 1) \\pi}{3} \\right\\} \\\\ \\sin \\pi & \\frac{1}{\\sqrt{3}} \\left\\{ -\\sin \\pi +\\sin \\frac{(3 \\pm 1) \\pi}{3} \\right\\} \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} -1 & 0 \\\\ 0 & \\pm 1 \\end{array} \\right) = \\left( \\begin{array}{cc} \\cos \\pi & \\mp \\sin \\pi \\\\ \\sin \\pi & \\pm \\cos \\pi \\end{array} \\right)\r\n\\end{align}\\]<\/li>\r\n

    5. 5*<\/strong>\u3000\\(k=4\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nA & = \\left( \\begin{array}{cc} \\cos \\frac{4 \\pi}{3} & \\frac{1}{\\sqrt{3}} \\left\\{ -\\cos \\frac{4 \\pi}{3} +\\cos \\frac{(4 \\pm 1) \\pi}{3} \\right\\} \\\\ \\sin \\frac{4 \\pi}{3} & \\frac{1}{\\sqrt{3}} \\left\\{ -\\sin \\frac{4 \\pi}{3} +\\sin \\frac{(4 \\pm 1) \\pi}{3} \\right\\} \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} -\\frac{1}{2} & \\pm \\frac{\\sqrt{3}}{2} \\\\ -\\frac{\\sqrt{3}}{2} & \\mp \\frac{1}{2} \\end{array} \\right) = \\left( \\begin{array}{cc} \\cos \\frac{4 \\pi}{3} & \\mp \\sin \\frac{4 \\pi}{3} \\\\ \\sin \\frac{4 \\pi}{3} & \\pm \\cos \\frac{4 \\pi}{3} \\end{array} \\right)\r\n\\end{align}\\]<\/li>\r\n

    6. 6*<\/strong>\u3000\\(k=5\\) \u306e\u3068\u304d\r\n\\[\\begin{align}\r\nA & = \\left( \\begin{array}{cc} \\cos \\frac{5 \\pi}{3} & \\frac{1}{\\sqrt{3}} \\left\\{ -\\cos \\frac{5 \\pi}{3} +\\cos \\frac{(5 \\pm 1) \\pi}{3} \\right\\} \\\\ \\sin \\frac{5 \\pi}{3} & \\frac{1}{\\sqrt{3}} \\left\\{ -\\sin \\frac{5 \\pi}{3} +\\sin \\frac{(5 \\pm 1) \\pi}{3} \\right\\} \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{cc} \\frac{1}{2} & \\mp \\frac{\\sqrt{3}}{2} \\\\ -\\frac{\\sqrt{3}}{2} & \\pm \\frac{1}{2} \\end{array} \\right) = \\left( \\begin{array}{cc} \\cos \\frac{5 \\pi}{3} & \\mp \\sin \\frac{5 \\pi}{3} \\\\ \\sin \\frac{5 \\pi}{3} & \\pm \\cos \\frac{5 \\pi}{3} \\end{array} \\right)\r\n\\end{align}\\]<\/li>\r\n<\/ol>\r\n

      1*<\/strong> \uff5e 6*<\/strong> \u3067\u6c42\u3081\u305f\u884c\u5217\u306f, \u539f\u70b9\u4e2d\u5fc3\u306e \\(\\dfrac{k \\pi}{3}\\) \u56de\u8ee2, \u3042\u308b\u3044\u306f\u3053\u308c\u3068 \\(x\\) \u8ef8\u5bfe\u79f0\u79fb\u52d5\u3092\u7d44\u5408\u305b\u305f\u79fb\u52d5\u3092\u8868\u3059\u306e\u3067, \u5404\u9802\u70b9\u306f\u3044\u305a\u308c\u304b\u306e\u9802\u70b9\u306b\u79fb\u52d5\u3059\u308b.
      \r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u884c\u5217\u306f\r\n\\[\\begin{align}\r\nA & = \\underline{\\left( \\begin{array}{cc} 1 & 0 \\\\ 0 & \\pm 1 \\end{array} \\right) , \\left( \\begin{array}{cc} -1 & 0 \\\\ 0 & \\pm 1 \\end{array} \\right) , } \\\\\r\n& \\quad \\underline{\\left( \\begin{array}{cc} \\frac{1}{2} & \\mp \\frac{\\sqrt{3}}{2} \\\\ \\frac{\\sqrt{3}}{2} & \\pm \\frac{1}{2} \\end{array} \\right) , \\left( \\begin{array}{cc} -\\frac{1}{2} & \\pm \\frac{\\sqrt{3}}{2} \\\\ \\frac{\\sqrt{3}}{2} & \\mp \\frac{1}{2} \\end{array} \\right) , } \\\\\r\n& \\qquad \\underline{\\left( \\begin{array}{cc} -\\frac{1}{2} & \\pm \\frac{\\sqrt{3}}{2} \\\\ -\\frac{\\sqrt{3}}{2} & \\mp \\frac{1}{2} \\end{array} \\right) , \\left( \\begin{array}{cc} \\frac{1}{2} & \\mp \\frac{\\sqrt{3}}{2} \\\\ -\\frac{\\sqrt{3}}{2} & \\pm \\frac{1}{2} \\end{array} \\right) \\quad ( \\text{\u8907\u53f7\u540c\u9806} )}\r\n\\end{align}\\]\r\n","protected":false},"excerpt":{"rendered":"\\(xy\\) \u5e73\u9762\u306b\u304a\u3044\u3066, \u539f\u70b9\u3092\u4e2d\u5fc3\u3068\u3057 P \\((1,0)\\) \u3092\u9802\u70b9\u306e \\(1\\) \u3064\u3068\u3059\u308b\u6b63 \\(6\\) \u89d2\u5f62\u3092 \\(X\\) \u3068\u3059\u308b. \\(A\\) \u3092 \\(2\\) \u6b21\u306e\u6b63\u65b9\u884c\u5217\u3068\u3057, \\(X\\) \u306e\u5404\u9802\u70b9 \\( … \u7d9a\u304d\u3092\u8aad\u3080 →<\/span><\/a>","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"inline_featured_image":false,"footnotes":""},"categories":[74],"tags":[148,14],"class_list":["post-501","post","type-post","status-publish","format-standard","hentry","category-tohoku_r_2010","tag-tohoku_r","tag-14"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/501","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/comments?post=501"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/501\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=501"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=501"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=501"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}