{"id":170,"date":"2011-12-02T22:20:32","date_gmt":"2011-12-02T13:20:32","guid":{"rendered":"http:\/\/roundown.main.jp\/nyushi\/?p=170"},"modified":"2021-03-18T10:14:42","modified_gmt":"2021-03-18T01:14:42","slug":"tkr200906","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/tkr200906\/","title":{"rendered":"\u6771\u5927\u7406\u7cfb2009\uff1a\u7b2c6\u554f"},"content":{"rendered":"
\n

\u5e73\u9762\u4e0a\u306e \\(2\\) \u70b9 P , Q \u306e\u8ddd\u96e2\u3092 \\(d( \\text{P} , \\text{Q} )\\) \u3068\u8868\u3059\u3053\u3068\u306b\u3059\u308b.\r\n\u5e73\u9762\u4e0a\u306b\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u4e00\u8fba\u306e\u9577\u3055\u304c \\(1000\\) \u306e\u6b63\u4e09\u89d2\u5f62 \\(\\triangle \\text{A} {} _ 1 \\text{A} {} _ 2 \\text{A} {} _ 3\\) \u304c\u3042\u308b.\r\n\\(\\triangle \\text{A} {} _ 1 \\text{A} {} _ 2 \\text{A} {} _ 3\\) \u306e\u5185\u90e8\u306b \\(3\\) \u70b9 \\(\\text{B} {} _ 1 , \\text{B} {} _ 2 , \\text{B} {} _ 3\\) \u3092,\r\n\\(d( \\text{A} {} _ n , \\text{B} {} _ n ) = 1 \\quad ( n=1, 2, 3 ) \\) \u3068\u306a\u308b\u3088\u3046\u306b\u3068\u308b. \u307e\u305f,\r\n\\[\\begin{align}\r\n\\overrightarrow{a _ 1} & = \\overrightarrow{\\text{A} {} _ 1 \\text{A} {} _ 2} ,\r\n\\quad \\overrightarrow{a _ 2} = \\overrightarrow{\\text{A} {} _ 2 \\text{A} {} _ 3} ,\r\n\\quad \\overrightarrow{a _ 3} = \\overrightarrow{\\text{A} {} _ 3 \\text{A} {} _ 1} , \\\\\r\n\\overrightarrow{e _ 1} & = \\overrightarrow{\\text{A} {} _ 1 \\text{B} {} _ 1} ,\r\n\\quad \\overrightarrow{e _ 2} = \\overrightarrow{\\text{A} {} _ 2 \\text{B} {} _ 2} ,\r\n\\quad \\overrightarrow{e _ 3} = \\overrightarrow{\\text{A} {} _ 3 \\text{B} {} _ 3}\r\n\\end{align}\\]\r\n\u3068\u304a\u304f. \\(n=1, 2, 3\\) \u306e\u305d\u308c\u305e\u308c\u306b\u5bfe\u3057\u3066, \u6642\u523b \\(0\\) \u306b \\(\\text{A} {} _ n\\) \u3092\u51fa\u767a\u3057,\r\n\\(\\overrightarrow{e _ n}\\) \u306e\u5411\u304d\u306b\u901f\u3055 \\(1\\) \u3067\u76f4\u9032\u3059\u308b\u70b9\u3092\u8003\u3048, \u6642\u523b \\(t\\) \u306b\u304a\u3051\u308b\u305d\u306e\u4f4d\u7f6e\u3092 \\(\\text{P} {} _ n (t)\\) \u3068\u8868\u3059\u3053\u3068\u306b\u3059\u308b.<\/p>\r\n

    \r\n

  1. (1)<\/strong>\u3000\u3042\u308b\u6642\u523b \\(t\\) \u3067 \\(d( \\text{P} {} _ 1 (t) , \\text{P} {} _ 2 (t) ) \\leqq 1\\) \u304c\u6210\u7acb\u3057\u305f.\r\n\u30d9\u30af\u30c8\u30eb \\(\\overrightarrow{e _ 1}-\\overrightarrow{e _ 2}\\) \u3068, \u30d9\u30af\u30c8\u30eb \\(\\overrightarrow{a _ 1}\\) \u3068\u306e\u306a\u3059\u89d2\u3092 \\(\\theta\\) \u3068\u304a\u304f.\r\n\u3053\u306e\u3068\u304d \\(| \\sin \\theta | \\leqq \\dfrac{1}{1000}\\) \u3068\u306a\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n

  2. (2)<\/strong>\u3000\u89d2\u5ea6 \\(\\theta _ 1 , \\theta _ 2 , \\theta _ 3\\) \u3092 \\(\\theta _ 1 = \\angle \\text{B} {} _ 1 \\text{A} {} _ 1 \\text{A} {} _ 2\\) ,\r\n\\(\\theta _ 2 = \\angle \\text{B} {} _ 2 \\text{A} {} _ 2 \\text{A} {} _ 3\\) ,\r\n\\(\\theta _ 3 = \\angle \\text{B} {} _ 3 \\text{A} {} _ 3 \\text{A} {} _ 1\\) \u306b\u3088\u3063\u3066\u5b9a\u7fa9\u3059\u308b.\r\n\\(\\alpha\\) \u3092 \\(0 \\lt \\alpha \\lt \\dfrac{\\pi}{2}\\) \u304b\u3064 \\(\\sin \\alpha =\\dfrac{1}{1000}\\) \u3092\u307f\u305f\u3059\u5b9f\u6570\u3068\u3059\u308b.\r\n(1)<\/strong> \u3068\u540c\u3058\u4eee\u5b9a\u306e\u3082\u3068\u3067, \\(\\theta _ 1 +\\theta _ 2\\) \u306e\u5024\u306e\u3068\u308b\u7bc4\u56f2\u3092 \\(\\alpha\\) \u3092\u7528\u3044\u3066\u8868\u305b.<\/p><\/li>\r\n

  3. (3)<\/strong>\u3000\u6642\u523b \\(t _ 1 , t _ 2 , t _ 3\\) \u306e\u305d\u308c\u305e\u308c\u306b\u304a\u3044\u3066, \u6b21\u304c\u6210\u7acb\u3057\u305f.\r\n\\[\r\nd(\\text{P} {} _ 2 (t _ 1) , \\text{P} {} _ 3 (t _ 1)) \\leqq 1 , \\quad d(\\text{P} {} _ 3 (t _ 2) , \\text{P} {} _ 1 (t _ 2)) \\leqq 1 ,\r\n\\quad d(\\text{P} {} _ 1 (t _ 3) , \\text{P} {} _ 2 (t _ 3)) \\leqq 1\r\n\\]\r\n\u3053\u306e\u3068\u304d, \u6642\u523b \\(T =\\dfrac{1000}{\\sqrt{3}}\\) \u306b\u304a\u3044\u3066\u540c\u6642\u306b\r\n\\[\r\nd(\\text{P} {} _ 1 (T) , \\text{O}) \\leqq 3 , \\quad d(\\text{P} {} _ 2 (T) , \\text{O}) \\leqq 3 , \\quad d(\\text{P} {} _ 3(T) , \\text{O}) \\leqq 3\r\n\\]\r\n\u304c\u6210\u7acb\u3059\u308b\u3053\u3068\u3092\u793a\u305b.<\/p><\/li>\r\n<\/ol>\r\n\"\"\r\n

    \r\n\r\n

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

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

    \\(d( \\text{P} {} _ 1(t) , \\text{P} {} _ 2(t) ) = \\left| t \\overrightarrow{e _ 1} -\\left( \\overrightarrow{e _ 2} -t \\overrightarrow{a _ 1} \\right) \\right|\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n& \\left| t \\overrightarrow{e _ 1} -\\left( \\overrightarrow{e _ 2} -t \\overrightarrow{a _ 1} \\right) \\right| \\\\\r\n& \\quad = t^2 \\left| \\overrightarrow{e _ 1} -\\overrightarrow{e _ 2} \\right|^2 -2t \\overrightarrow{a _ 1} \\cdot \\left( \\overrightarrow{e _ 1} - \\overrightarrow{e _ 2} \\right) +\\left| \\overrightarrow{a _ 1} \\right|^2 \\\\\r\n& \\quad = t^2 \\left| \\overrightarrow{e _ 1} -\\overrightarrow{e _ 2} \\left|^2 -2t \\right| \\overrightarrow{a _ 1} \\right| \\left| \\overrightarrow{e _ 1} - \\overrightarrow{e _ 2} \\right| \\cos \\theta +\\left| \\overrightarrow{a _ 1} \\right|^2 \\\\\r\n& \\quad = t^2 \\left| \\overrightarrow{e _ 1} -\\overrightarrow{e _ 2} \\left|^2 -2000t \\right| \\overrightarrow{e _ 1} - \\overrightarrow{e _ 2} \\right| \\cos \\theta +1000^2 \\leqq 1\\\\\r\n\\text{\u2234} \\quad & t^2 \\left| \\overrightarrow{e _ 1} -\\overrightarrow{e _ 2} \\left|^2 -2000t \\right| \\overrightarrow{e _ 1} - \\overrightarrow{e _ 2} \\right| \\cos \\theta +1000^2-1 \\leqq 0\r\n\\end{align}\\]\r\n\u3053\u308c\u304c\u5b9f\u6570\u89e3 \\(t\\) \u3092\u3082\u3064\u306e\u3067, \u5224\u5225\u5f0f \\(D\\) \u306b\u3064\u3044\u3066\r\n\\[\\begin{gather}\r\n\\dfrac{D}{4} = 1000^2 \\left| \\overrightarrow{e _ 1} -\\overrightarrow{e _ 2} \\right|^2 \\cos^2 \\theta -( 1000^2-1 ) \\left| \\overrightarrow{e _ 1} -\\overrightarrow{e _ 2} \\right|^2 \\geqq 0 \\\\\r\n1000^2 \\left( 1 -\\sin^2 \\theta \\right) -( 1000^2-1 ) \\geqq 0 \\\\\r\n\\sin^2 \\theta \\leqq \\dfrac{1}{1000^2} \\\\\r\n\\text{\u2234} \\quad \\underline{\\left| \\sin \\theta \\right| \\leqq \\dfrac{1}{1000}}\r\n\\end{gather}\\]\r\n

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

    \\(\\overrightarrow{e _ 1} -\\overrightarrow{e _ 2}\\) \u3068 \\(\\overrightarrow{a _ 1}\\) \u306e\u306a\u3059\u89d2 \\(\\theta\\) \u306b\u3064\u3044\u3066\u8003\u3048\u308b\u3068\r\n\\[\r\n\\left| \\dfrac{\\theta _ 1 +\\left( \\dfrac{\\pi}{3} -\\theta _ 2 \\right)}{2} -\\theta _ 1 \\right| = \\left| \\dfrac{\\pi}{6} -\\dfrac{\\theta _ 1 +\\theta _ 2}{2} \\right|\r\n\\]\r\n\u3053\u308c\u3068 (1)<\/strong> \u306e\u7d50\u679c\u3088\u308a\r\n\\[\\begin{gather}\r\n\\left| \\dfrac{\\pi}{6} -\\dfrac{\\theta _ 1 +\\theta _ 2}{2} \\right| \\leqq \\alpha \\\\\r\n-2 \\alpha \\leqq \\dfrac{\\pi}{3} -\\left( \\theta _ 1 +\\theta _ 2 \\right) \\leqq 2\\alpha \\\\\r\n\\text{\u2234} \\quad \\underline{\\dfrac{\\pi}{3} -2\\alpha \\leqq \\theta _ 1 +\\theta _ 2 \\leqq \\dfrac{\\pi}{3} +2\\alpha}\r\n\\end{gather}\\]\r\n

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

    (2)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070,\r\n\\[\r\n\\left\\{ \\begin{array}{l} \\dfrac{\\pi}{3} -2\\alpha \\leqq \\theta _ 1 +\\theta _ 2 \\leqq \\dfrac{\\pi}{3} +2\\alpha \\\\ \\dfrac{\\pi}{3} -2\\alpha \\leqq \\theta _ 2 +\\theta _ 3 \\leqq \\dfrac{\\pi}{3} +2\\alpha \\\\ \\dfrac{\\pi}{3} -2\\alpha \\leqq \\theta _ 3 +\\theta _ 1 \\leqq \\dfrac{\\pi}{3} +2\\alpha \\end{array} \\right.\r\n\\]\r\n\u3053\u3053\u3067, \\({\\theta _ i}' =\\theta _ i -\\dfrac{\\pi}{6}\\) \uff08 \\(i = 1, 2, 3\\) \uff09\u3068\u304a\u3051\u3070\r\n\\[\r\n\\left\\{ \\begin{array}{ll} -2\\alpha \\leqq {\\theta _ 1}' +{\\theta _ 2}' \\leqq 2\\alpha & \\ \\text{...[1]} \\\\ -2\\alpha \\leqq {\\theta _ 2}' +{\\theta _ 3}' \\leqq 2\\alpha & \\ \\text{...[2]} \\\\ -2\\alpha \\leqq {\\theta _ 3}' +{\\theta _ 1}' \\leqq 2\\alpha & \\ \\text{...[3]} \\end{array} \\right.\r\n\\]\r\n\\([1]+[3]-[2]\\) \u3088\u308a\r\n\\[\\begin{align}\r\n-6 \\alpha \\leqq 2{\\theta _ 1}' & \\leqq 6\\alpha \\\\\r\n\\text{\u2234} \\quad -3\\alpha \\leqq {\\theta _ 1}' & \\leqq 3 \\alpha \\quad ... [4]\r\n\\end{align}\\]\r\n\\(\\text{OP} {} _ 1 = \\text{OA} {} _ 1 = T\\) \u306a\u306e\u3067\r\n\\[\\begin{align}\r\n0 \\leqq d(\\text{P} {} _ 1(T) , \\text{O}) & = 2T \\sin \\left| \\dfrac{{\\theta _ 1}'}{2} \\right| \\\\\r\n& \\leqq 2T \\sin \\dfrac{3\\alpha}{2} \\quad ( \\ \\text{\u2235} \\ [4] \\ ) \\\\\r\n& \\lt 4T \\sin \\alpha \\cos \\alpha \\\\\r\n& \\leqq \\dfrac{4}{\\sqrt{3}} \\cos \\alpha \\\\\r\n& \\lt \\dfrac{4}{\\sqrt{3}} \\lt 3 \\quad ( \\ \\text{\u2235} \\ 4^2 \\lt 3^3 \\ ) \\\\\r\n\\text{\u2234} \\quad & d(\\text{P} {} _ 1(T) , \\text{O}) \\leqq 3\r\n\\end{align}\\]\r\n\u3053\u308c\u3068\u540c\u69d8\u306b\u3059\u308c\u3070,\r\n\\[\r\nd(\\text{P} {} _ 2(T) , \\text{O}) \\leqq 3 , \\quad d(\\text{P} {} _ 3(T) , \\text{O}) \\leqq 3\r\n\\]\r\n\u3088\u3063\u3066, \u984c\u610f\u306f\u793a\u3055\u308c\u305f.<\/p>\r\n","protected":false},"excerpt":{"rendered":"\u5e73\u9762\u4e0a\u306e \\(2\\) \u70b9 P , Q \u306e\u8ddd\u96e2\u3092 \\(d( \\text{P} , \\text{Q} )\\) \u3068\u8868\u3059\u3053\u3068\u306b\u3059\u308b. \u5e73\u9762\u4e0a\u306b\u70b9 O \u3092\u4e2d\u5fc3\u3068\u3059\u308b\u4e00\u8fba\u306e\u9577\u3055\u304c \\(1000\\) \u306e\u6b63\u4e09\u89d2\u5f62 \\(\\triangle … \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":[18],"tags":[139,15],"class_list":["post-170","post","type-post","status-publish","format-standard","hentry","category-tokyo_r_2009","tag-tokyo_r","tag-15"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/170","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=170"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/170\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=170"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=170"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=170"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}