{"id":446,"date":"2012-10-29T23:17:36","date_gmt":"2012-10-29T14:17:36","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=446"},"modified":"2021-09-16T05:57:09","modified_gmt":"2021-09-15T20:57:09","slug":"ngr201001","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/ngr201001\/","title":{"rendered":"\u540d\u53e4\u5c4b\u5927\u7406\u7cfb2010\uff1a\u7b2c1\u554f"},"content":{"rendered":"
\n

\u5ea7\u6a19\u7a7a\u9593\u306b \\(8\\) \u70b9\r\n\\[\\begin{align}\r\n& \\text{O} \\ (0,0,0) , \\quad \\text{P} \\ (1,0,0) , \\quad \\text{Q} \\ (1,1,0) , \\quad \\text{R} \\ (0,1,0) , \\\\\r\n& \\text{A} \\ (0,0,1) , \\quad \\text{B} \\ (1,0,1) , \\quad \\text{C} \\ (1,1,1) , \\quad \\text{D} \\ (0,1,1)\n\\end{align}\\]\r\n\u3092\u3068\u308a, \u7dda\u5206 BC \u306e\u4e2d\u70b9\u3092 M \u3068\u3059\u308b. \u7dda\u5206 RD \u4e0a\u306e\u70b9\u3092 N \\((0,1,t)\\) \u3068\u3057, \\(3\\) \u70b9 O , M , N \u3092\u901a\u308b\u5e73\u9762\u3068\u7dda\u5206 PD \u304a\u3088\u3073\u7dda\u5206 PB \u3068\u306e\u4ea4\u70b9\u3092\u305d\u308c\u305e\u308c K , L \u3068\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000K \u306e\u5ea7\u6a19\u3092 \\(t\\) \u3067\u8868\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u56db\u9762\u4f53 OKLP \u306e\u4f53\u7a4d\u3092 \\(V(t)\\) \u3068\u3059\u308b. N \u304c\u7dda\u5206 RD \u4e0a\u3092 R \u304b\u3089 D \u307e\u3067\u52d5\u304f\u3068\u304d, \\(V(t)\\) \u306e\u6700\u5927\u5024\u3068\u6700\u5c0f\u5024\u304a\u3088\u3073\u305d\u308c\u3089\u3092\u4e0e\u3048\u308b \\(t\\) \u306e\u5024\u3092\u305d\u308c\u305e\u308c\u6c42\u3081\u3088.<\/p><\/li>\r\n<\/ol>\r\n

    \r\n\r\n

    \u3010 \u89e3 \u7b54 \u3011<\/h4>\r\n\"\"\r\n

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

    O , M , N \u3092\u901a\u308b\u5e73\u9762\u3092 \\(\\alpha\\) \u3068\u304a\u304f.
    \r\n\u70b9 K \u306f \\(\\alpha\\) \u4e0a\u306b\u3042\u308b\u306e\u3067, \u5b9f\u6570 \\(u , v\\) \u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OK}} & = u \\overrightarrow{\\text{OM}} +v \\overrightarrow{\\text{ON}} \\\\\r\n& = u \\left( \\begin{array}{c} 1 \\\\ \\dfrac{1}{2} \\\\ 1 \\end{array} \\right) +v \\left( \\begin{array}{c} 0 \\\\ 1 \\\\ t\r\n\\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{c} u \\\\ \\dfrac{u}{2} +v \\\\ u+tv \\end{array} \\right) \\quad ... [1]\r\n\\end{align}\\]\r\n\u3068\u8868\u305b\u308b.
    \r\n\u307e\u305f, \u70b9 K \u306f\u7dda\u5206 PD \u4e0a\u306b\u3042\u308b\u306e\u3067, \u5b9f\u6570 \\(s \\ ( 0 \\leqq s \\leqq 1 )\\) \u3092\u7528\u3044\u3066\r\n\\[\\begin{align}\r\n\\overrightarrow{\\text{OK}} & = (1-s) \\overrightarrow{\\text{OP}} +s \\overrightarrow{\\text{OD}} \\\\\r\n& = (1-s) \\left( \\begin{array}{c} 1 \\\\ 0 \\\\ 0 \\end{array} \\right) +s \\left( \\begin{array}{c} 0 \\\\ 1 \\\\ 1 \\end{array} \\right) \\\\\r\n& = \\left( \\begin{array}{c} 1-s \\\\ s \\\\ s \\end{array} \\right) \\quad ... [2]\r\n\\end{align}\\]\r\n\u3068\u8868\u305b\u308b.
    \r\n[1] [2] \u3088\u308a\r\n\\[\r\nu = 1-s , \\ \\dfrac{u}{2} +v = s , \\ u+tv = s\n\\]\r\n\u3053\u308c\u3092\u89e3\u304f\u3068\r\n\\[\r\nu = \\dfrac{2 (1-t)}{4-3t} , \\ v = \\dfrac{1}{4-3t} , \\ s = \\dfrac{2-t}{4-3t}\n\\]\r\n\u3053\u3053\u3067 \\(0 \\leqq t \\leqq 1\\) \u306a\u306e\u3067, \\(0 \\leqq s \\leqq 1\\) \u3092\u6e80\u305f\u3057\u3066\u3044\u308b.
    \r\n\u3088\u3063\u3066\r\n\\[\r\n\\text{K} \\ \\underline{\\left( \\dfrac{2 (1-t)}{4-3t} , \\dfrac{2-t}{4-3t} , \\dfrac{2-t}{4-3t} \\right)}\n\\]\r\n

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

    \u70b9 L \u306f \\(\\alpha\\) \u4e0a\u306b\u3042\u308b\u306e\u3067, (1)<\/strong> \u3068\u540c\u69d8\u306b\r\n\\[\r\n\\overrightarrow{\\text{OL}} = \\left( \\begin{array}{c} u \\\\ \\dfrac{u}{2} +v \\\\ u+tv \\end{array} \\right) \\quad ... [3]\r\n\\]\r\n\u3068\u8868\u305b\u308b.
    \r\n\u307e\u305f, \u70b9 L \u306f\u7dda\u5206 PA \u4e0a\u306b\u3042\u308b\u306e\u3067, \u5b9f\u6570 \\(\\ell\\) \uff08 \\(0 \\leqq \\ell \\leqq 1\\) \uff09\u3092\u7528\u3044\u3066\r\n\\[\r\n\\overrightarrow{\\text{OL}} = \\left( \\begin{array}{c} 1 \\\\ 0 \\\\ \\ell \\end{array} \\right) \\quad ... [4]\r\n\\]\r\n\u3068\u8868\u305b\u308b.
    \r\n[3] [4] \u3088\u308a\r\n\\[\r\nu = 1 , \\ \\dfrac{u}{2} +v =0 , \\ u+tv = \\ell\n\\]\r\n\u3053\u308c\u3092\u89e3\u304f\u3068\r\n\\[\r\nu = 1 , \\ v= -\\dfrac{1}{2} , \\ \\ell = \\dfrac{2-t}{2}\n\\]\r\n\u3057\u305f\u304c\u3063\u3066\r\n\\[\r\n\\text{L} \\ \\left( 1 , 0 , \\dfrac{2-t}{2} \\right)\n\\]\r\n\u56db\u9762\u4f53 OKLP \u306b\u3064\u3044\u3066, \u25b3OPL \u3092\u5e95\u9762\u3068\u307f\u306a\u305b\u3070\r\n\\[\\begin{align}\r\nV(t) & = \\dfrac{1}{3} \\left( \\dfrac{1}{2} \\cdot 1 \\cdot \\dfrac{2-t}{2} \\right) \\cdot \\dfrac{2-t}{4-3t} \\\\\r\n& = \\dfrac{1}{12} \\cdot \\underline{\\dfrac{(2-t)^2}{4-3t}} _ {[\\text{A}]}\n\\end{align}\\]\r\n[A] \u3092 \\(f(t)\\) \u3068\u304a\u3044\u3066, \\(0 \\leqq t \\leqq 1\\) \u306b\u304a\u3051\u308b\u5897\u6e1b\u3092\u8abf\u3079\u308c\u3070\u3088\u3044.\r\n\\[\\begin{align}\r\nf'(t) & = \\dfrac{-2(2-t)(4-3t) +3(2-t)^2}{(4-3t)^2} \\\\\r\n& = \\dfrac{(3t-2)(2-t)}{(4-3t)^2}\n\\end{align}\\]\r\n\u3057\u305f\u304c\u3063\u3066, \\(0 \\leqq t \\leqq 1\\) \u306b\u304a\u3051\u308b \\(f(t)\\) \u306e\u5897\u6e1b\u306f\u4e0b\u8868\u306e\u3088\u3046\u306b\u306a\u308b.\r\n\\[\r\n\\begin{array}{c|ccccc} t & 0 & \\cdots & \\dfrac{2}{3} & \\cdots & 1 \\\\ \\hline f'(t) & & - & 0 & + & \\\\ \\hline f(t) & 1 & \\searrow & \\dfrac{8}{9} & \\nearrow & 1 \\end{array}\r\n\\]\r\n\u3088\u3063\u3066, \\(V(t)\\) \u306b\u3064\u3044\u3066<\/p>\r\n