{"id":1123,"date":"2015-06-22T09:38:31","date_gmt":"2015-06-22T00:38:31","guid":{"rendered":"http:\/\/www.roundown.net\/nyushi\/?p=1123"},"modified":"2021-09-09T06:40:00","modified_gmt":"2021-09-08T21:40:00","slug":"osr201404","status":"publish","type":"post","link":"https:\/\/www.roundown.net\/nyushi\/osr201404\/","title":{"rendered":"\u962a\u5927\u7406\u7cfb2014\uff1a\u7b2c4\u554f"},"content":{"rendered":"
\n

\u534a\u5f84 \\(1\\) \u306e \\(2\\) \u3064\u306e\u7403 \\(S _ 1\\) \u3068 \\(S _ 2\\) \u304c \\(1\\) \u70b9\u3067\u63a5\u3057\u3066\u3044\u308b.\r\n\u4e92\u3044\u306b\u91cd\u306a\u308b\u90e8\u5206\u306e\u306a\u3044\u7b49\u3057\u3044\u534a\u5f84\u3092\u6301\u3064 \\(n\\) \u500b\uff08 \\(n \\geqq 3\\) \uff09\u306e\u7403 \\(T _ 1 , T _ 2 , \\cdots , T _ n\\) \u304c\u3042\u308a, \u6b21\u306e\u6761\u4ef6 (\u30a2) (\u30a4) \u3092\u6e80\u305f\u3059.<\/p>\r\n

    \r\n

  1. (\u30a2)\u3000\\(T _ {i}\\) \u306f \\(S _ 1 , S _ 2\\) \u306b\u305d\u308c\u305e\u308c \\(1\\) \u70b9\u3067\u63a5\u3057\u3066\u3044\u308b\uff08 \\(i = 1 , 2 , \\cdots , n\\) \uff09.<\/p><\/li>\r\n

  2. (\u30a4)\u3000\\(T _ i\\) \u306f \\(T _ {i+1}\\) \u306b \\(1\\) \u70b9\u3067\u63a5\u3057\u3066\u304a\u308a\uff08 \\(i = 1 , 2 , \\cdots , n-1\\) \uff09, \u305d\u3057\u3066 \\(T _ n\\) \u306f \\(T _ 1\\) \u306b \\(1\\) \u70b9\u3067\u63a5\u3057\u3066\u3044\u308b.<\/p><\/li>\r\n<\/ol>\r\n

    \u3053\u306e\u3068\u304d, \u4ee5\u4e0b\u306e\u554f\u3044\u306b\u7b54\u3048\u3088.<\/p>\r\n

      \r\n

    1. (1)<\/strong>\u3000\\(T _ 1 , T _ 2 , \\cdots , T _ n\\) \u306e\u5171\u901a\u306e\u534a\u5f84 \\(r _ n\\) \u3092\u6c42\u3081\u3088.<\/p><\/li>\r\n

    2. (2)<\/strong>\u3000\\(S _ 1\\) \u3068 \\(S _ 2\\) \u306e\u4e2d\u5fc3\u3092\u7d50\u3076\u76f4\u7dda\u306e\u307e\u308f\u308a\u306b \\(T _ 1\\) \u3092\u56de\u8ee2\u3057\u3066\u3067\u304d\u308b\u56de\u8ee2\u4f53\u306e\u4f53\u7a4d\u3092 \\(V _ n\\) \u3068\u3057, \\(T _ 1 , T _ 2 , \\cdots , T _ n\\) \u306e\u4f53\u7a4d\u306e\u548c\u3092 \\(W _ n\\) \u3068\u3059\u308b\u3068\u304d, \u6975\u9650\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{W _ n}{V _ n} \\ .\r\n\\]\r\n\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

      \\(S _ 1 , T _ 1 , T _ 2\\) \u306e\u4e2d\u5fc3\u3092\u305d\u308c\u305e\u308c O , P , Q , \\(S _ 1\\) \u3068 \\(S _ 2\\) , \\(T _ 1\\) \u3068 \\(T _ 2\\) \u306e\u63a5\u70b9\u3092\u305d\u308c\u305e\u308c R , S \u3068\u304a\u304f.
      \r\n\u3053\u306e\u3068\u304d, \\(\\text{OP} = 1+r _ n\\) \u3067\u3042\u308a, \\(\\text{PR} = d _ n\\) \u3068\u304a\u304f\u3068\r\n\\[\r\nd _ n = \\sqrt{(1+r _ n)^2 -1^2} = \\sqrt{r _ n ( r _ n +2 )} \\quad ... [1] \\ .\r\n\\]\r\n\u25b3PRS \u306b\u7740\u76ee\u3059\u308c\u3070\r\n\\[\r\n\\sin \\dfrac{\\pi}{n} = \\dfrac{r _ n}{d _ n} = \\sqrt{\\dfrac{r _ n}{r _ n +2}} \\ .\r\n\\]\r\n\u4e21\u8fba\u3092\u5e73\u65b9\u3059\u308c\u3070\r\n\\[\\begin{align}\r\n\\dfrac{r _ n}{r _ n +2} & = \\sin^2 \\dfrac{\\pi}{n} \\\\\r\n\\left( 1 -\\sin^2 \\dfrac{\\pi}{n} \\right) r _ n & = \\sin^2 \\dfrac{\\pi}{n} \\\\\r\n\\text{\u2234} \\quad r _ n & = \\underline{2 \\tan^2 \\dfrac{\\pi}{n}} \\ .\r\n\\end{align}\\]\r\n

      (2)<\/strong>\r\n\\[\r\nW _ n = \\dfrac{4n \\pi}{3} {r _ n}^3 \\quad ... [2] \\ .\r\n\\]\r\n\\(T _ 1\\) \u306e\u56de\u8ee2\u4f53\u306f, \u4e2d\u5fc3 \\(( 0 , d _ n )\\) , \u534a\u5f84 \\(r _ n\\) \u306e\u5186 \\(C _ n\\) \u3092 \\(x\\) \u8ef8\u306e\u307e\u308f\u308a\u306e\u56de\u8ee2\u4f53\u3067\u3042\u308b.
      \r\n\\(C _ n\\) \u306e\u5f0f\u306b\u3064\u3044\u3066\r\n\\[\\begin{align}\r\nx^2 & + \\left( y -d _ n \\right)^2 = {r _ n}^2 \\\\\r\n\\text{\u2234} \\quad y & = d _ n \\pm \\sqrt{{r _ n}^2 -y^2} \\quad ... [3] \\ .\r\n\\end{align}\\]\r\n[3] \u306e\u8907\u53f7\u306e\u3046\u3061, \u6b63, \u8ca0\u3092\u3068\u308b\u3068\u304d\u306e\u5f0f\u3092\u305d\u308c\u305e\u308c \\(y _ {+} , y _ {-}\\) \u3068\u304a\u304f\u3068\r\n\\[\\begin{align}\r\nV _ n & = 2 \\pi \\displaystyle\\int _ 0^{r _ n} \\left( {y _ {+}}^2 -{y _ {-}}^2 \\right) \\, dx \\\\\r\n& = 2 \\pi \\cdot 4 d _ n \\underline{\\displaystyle\\int _ 0^{r _ n} \\sqrt{{r _ n}^2 -y^2} \\, dx} _ {[4]} \\\\\r\n& = 8 \\pi d _ n \\cdot \\dfrac{\\pi {r _ n}^2}{4} \\\\\r\n& = 2 \\pi^2 d _ n {r _ n}^2 \\quad ... [5] \\ .\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \u4e0b\u7dda\u90e8 [4] \u306f, \u534a\u5f84 \\(r _ n\\) \u306e\u56db\u5206\u5186\u306e\u9762\u7a4d\u3092\u8868\u3059\u3053\u3068\u3092\u7528\u3044\u305f.
      \r\n\u3057\u305f\u304c\u3063\u3066, [2] [5] \u3088\u308a\r\n\\[\\begin{align}\r\n\\dfrac{W _ n}{V _ n} & = \\dfrac{2n r _ n}{3 \\pi d _ n} \\\\\r\n& = \\dfrac{2}{3 \\pi} \\sqrt{\\underline{\\dfrac{n^2 r _ n}{r _ n +2}} _ {[6]}} \\quad ( \\ \\text{\u2235} \\ [1] \\ ) \\ .\r\n\\end{align}\\]\r\n\u3053\u3053\u3067, \u4e0b\u7dda\u90e8 [6] \u306b\u3064\u3044\u3066, (1)<\/strong> \u306e\u7d50\u679c\u3092\u7528\u3044\u308c\u3070\r\n\\[\\begin{align}\r\n[6] & = \\dfrac{n^2 \\tan^2 \\frac{\\pi}{n}}{1 +\\tan^2 \\frac{\\pi}{n}} \\\\\r\n& = n^2 \\cdot \\sin^2 \\dfrac{\\pi}{n} \\\\\r\n& = \\pi^2 \\cdot \\left( \\dfrac{\\sin \\frac{\\pi}{n}}{\\frac{\\pi}{n}} \\right)^2 \\\\\r\n& \\rightarrow \\pi^2 \\quad ( \\ n \\rightarrow \\infty \\text{\u306e\u3068\u304d} \\ \\ ) \\ .\r\n\\end{align}\\]\r\n\u3088\u3063\u3066, \u6c42\u3081\u308b\u6975\u9650\u5024\u306f\r\n\\[\r\n\\displaystyle\\lim _ {n \\rightarrow \\infty} \\dfrac{W _ n}{V _ n} = \\dfrac{2}{3 \\pi} \\cdot \\pi = \\underline{\\dfrac{2}{3}} \\ .\r\n\\]\r\n","protected":false},"excerpt":{"rendered":"\u534a\u5f84 \\(1\\) \u306e \\(2\\) \u3064\u306e\u7403 \\(S _ 1\\) \u3068 \\(S _ 2\\) \u304c \\(1\\) \u70b9\u3067\u63a5\u3057\u3066\u3044\u308b. \u4e92\u3044\u306b\u91cd\u306a\u308b\u90e8\u5206\u306e\u306a\u3044\u7b49\u3057\u3044\u534a\u5f84\u3092\u6301\u3064 \\(n\\) \u500b\uff08 \\(n \\geqq 3\\) \uff09\u306e\u7403 \\(T … \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":[115],"tags":[142,112],"class_list":["post-1123","post","type-post","status-publish","format-standard","hentry","category-osaka_r_2014","tag-osaka_r","tag-112"],"_links":{"self":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1123","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=1123"}],"version-history":[{"count":0,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/posts\/1123\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/media?parent=1123"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/categories?post=1123"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.roundown.net\/nyushi\/wp-json\/wp\/v2\/tags?post=1123"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}