{"id":6827,"date":"2020-10-24T20:03:23","date_gmt":"2020-10-24T11:03:23","guid":{"rendered":"https:\/\/pandanote.info\/?p=6827"},"modified":"2022-08-07T12:55:51","modified_gmt":"2022-08-07T03:55:51","slug":"%e7%ac%ac2%e7%a8%ae%e6%a5%95%e5%86%86%e7%a9%8d%e5%88%86%e3%81%ae%e5%b0%8e%e5%87%ba%e3%81%ae%e6%96%b9%e6%b3%95%e3%81%8c%e3%81%86%e3%82%8d%e8%a6%9a%e3%81%88%e3%81%a0%e3%81%a8%e3%80%81%e3%81%84%e3%81%96","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=6827","title":{"rendered":"\u7b2c2\u7a2e\u6955\u5186\u7a4d\u5206\u306e\u5c0e\u51fa\u306e\u65b9\u6cd5\u304c\u3046\u308d\u899a\u3048\u3060\u3068\u3001\u3044\u3056\u3068\u3044\u3046\u3068\u304d\u306b\u30cf\u30de\u308b\u4ef6"},"content":{"rendered":"

\u306f\u3058\u3081\u306b<\/h2>\n

\u7d50\u8ad6\u306b\u3064\u3044\u3066\u306fWikipedia<\/a>\u3092\u898b\u3066\u3044\u305f\u3060\u304f\u306e\u304c\u4e00\u756a\u65e9\u3044\u306e\u3067\u3059\u304c\u3001<\/p>\n

\u300c\u306a\u3093\u3067\u305d\u3046\u306a\u308b\u3093\u3060\u3063\u3051?\u300d<\/p>\n

\u306a\u3069\u3069\u8003\u3048\u59cb\u3081\u3066\u8a08\u7b97\u3092\u9032\u3081\u308b\u3068\u3001\u8a08\u7b97\u306e\u7d50\u679c\u306e\u5f0f\u3092\u898b\u3066\u7126\u308b\u3053\u3068\u306b\u306a\u308b\u306e\u3067\u3001\u30e1\u30e2\u3063\u3066\u304a\u304f\u3053\u3068\u306b\u3057\u307e\u3057\u305f\u3002<\/p>\n

\u30b5\u30af\u30b5\u30af\u3068\u8a08\u7b97<\/h2>\n

\u76f4\u622a\u306a\u5c0e\u51fa\u6cd5<\/h3>\n

\u6700\u521d\u306b\u6700\u3082\u76f4\u622a(\u3068\u601d\u308f\u308c\u308b)\u65b9\u6cd5\u3067\u4ee5\u4e0b\u306e\u5f0f\u306b\u305f\u3069\u308a\u7740\u304f\u3053\u3068\u3092\u8003\u3048\u307e\u3059(Wikipedia\u306b\u8a18\u8f09\u306e\u5f0f\u3067\u3059)\u3002
\n\\begin{align}
\nE(\\varphi, k) &= \\int_0^{\\varphi}\\sqrt{1-k^2\\sin^2\\theta}\\,d\\theta \\label{eq:secondellipticintegral}
\n\\end{align}<\/p>\n

\u307e\u305a\u3001$xy$\u5e73\u9762\u4e0a\u306e\u6955\u5186
\n\\begin{align}
\n \\frac{x^2}{a^2}+\\displaystyle\\frac{y^2}{b^2} &= 1 \\quad (a > b > 0) \\label{eq:ellipse}
\n\\end{align}
\n(\u4e0b\u56f3\u53c2\u7167)\u306e\u5468\u306e\u9577\u3055\u3092\u6c42\u3081\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n

\"\"<\/a><\/p>\n

\u306a\u304a\u3001\u3053\u306e\u6955\u5186\u306f$x$\u8ef8\u53ca\u3073$y$\u8ef8\u306b\u95a2\u3057\u3066\u5bfe\u79f0\u306a\u306e\u3067\u3001$x \\ge 0, y \\ge 0$\u306e\u90e8\u5206(\u4e0b\u56f3\u306e\u9752\u3044\u5b9f\u7dda\u306e\u90e8\u5206)\u3060\u3051\u8003\u3048\u307e\u3059\u3002\u6955\u5186\u306e1\u5468\u5206\u306e\u9577\u3055\u304c\u5fc5\u8981\u3067\u3042\u308c\u3070$x \\ge 0, y \\ge 0$\u306e\u90e8\u5206\u306e\u5f27\u306e\u9577\u3055\u3092\u6c42\u3081\u3001\u305d\u308c\u30924\u500d\u3059\u308c\u3070\u3088\u3044\u3068\u3044\u3046\u5bf8\u6cd5\u3067\u3059\u3002<\/p>\n

\"\"<\/a><\/p>\n

\u3053\u3053\u3067\u3001$y$\u8ef8\u306e\u6b63\u306e\u5411\u304d\u306e\u534a\u76f4\u7dda\u3068\u306e\u306a\u3059\u89d2(\u53f3\u56de\u308a\u3092\u6b63\u3068\u3057\u307e\u3059\u3002\u2190\u3053\u3053\u91cd\u8981)\u304c$\\theta$\u3067\u3042\u308a\u3001\u539f\u70b9$O$\u3092\u7aef\u70b9\u3068\u3059\u308b\u534a\u76f4\u7dda\u3092\u8003\u3048\u3001\u305d\u308c\u3068\u6955\u5186(\\ref{eq:ellipse})\u306e\u4ea4\u70b9\u3092$P$\u3068\u3057\u307e\u3059\u3002<\/p>\n

\u307e\u305f\u3001$y$\u8ef8\u306e\u6b63\u306e\u5411\u304d\u306e\u534a\u76f4\u7dda\u3068\u306e\u306a\u3059\u89d2\u304c$\\theta+d\\theta$\u3067\u3042\u308a\u3001\u539f\u70b9$O$\u3092\u7aef\u70b9\u3068\u3059\u308b\u534a\u76f4\u7dda\u3092\u8003\u3048\u3001\u305d\u308c\u3068\u6955\u5186(\\ref{eq:ellipse})\u306e\u4ea4\u70b9\u3092$Q$\u3068\u3057\u307e\u3059\u3002<\/p>\n

$P$,$Q$\u306e\u4f4d\u7f6e\u95a2\u4fc2\u3092\u4e0b\u56f3\u306b\u793a\u3057\u307e\u3059\u3002<\/p>\n

\"\"<\/a><\/p>\n

\u3059\u308b\u3068\u3001$P$\u306e\u5ea7\u6a19\u306f$(a\\sin\\theta,b\\cos\\theta)$\u3001$Q$\u306e\u5ea7\u6a19\u306f$(a\\sin(\\theta+d\\theta),b\\cos(\\theta+d\\theta))$\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u3053\u3053\u3067\u3001$d\\theta$\u304c$\\theta$\u3068\u6bd4\u8f03\u3057\u3066\u5fae\u5c0f\u306a\u6b63\u306e\u6570\u3067\u3042\u308b\u3068\u3057\u3001\u5fae\u5c0f\u306a\u9577\u3055\u306e\u7dda\u5206$PQ$\u306e\u9577\u3055$|PQ|$\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002<\/p>\n

\u3059\u308b\u3068\u2026
\n\\begin{align}
\n |PQ| &= \\sqrt{a^2\\{\\sin(\\theta+d\\theta)-\\sin\\theta\\}^2+b^2\\{\\cos(\\theta+d\\theta)-\\cos\\theta\\}^2} \\nonumber \\\\
\n &= \\sqrt{a^2\\{\\sin\\theta\\cos d\\theta+\\cos\\theta\\sin d\\theta-\\sin\\theta\\}^2+b^2\\{\\cos\\theta\\cos d\\theta-\\sin\\theta\\sin d\\theta-\\cos\\theta\\}^2}\\label{eq:dsfirst}
\n\\end{align}
\n\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u3053\u3053\u3067\u3001(\\ref{eq:dsfirst})\u5f0f\u3092$d\\theta$\u304c\u5fae\u5c0f\u3067\u3042\u308b\u3053\u3068\u306b\u7740\u76ee\u3057\u3066\u3055\u3089\u306b\u5909\u5f62\u3059\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n

$d\\theta$\u306f\u5fae\u5c0f\u3067\u3042\u308b\u305f\u3081\u3001$\\sin d\\theta = d\\theta, \\cos d\\theta = 1$\u3067\u3042\u308b\u3068\u8003\u3048\u307e\u3059\u3002<\/p>\n

\u3059\u308b\u3068\u2026
\n\\begin{align}
\n |PQ| &= \\sqrt{a^2\\{\\sin\\theta+d\\theta\\cos\\theta-\\sin\\theta\\}^2+b^2\\{\\cos\\theta-d\\theta\\sin\\theta-\\cos\\theta\\}^2} \\nonumber \\cr
\n &= \\sqrt{a^2\\{d\\theta\\cos\\theta\\}^2+b^2\\{-d\\theta\\sin\\theta\\}^2} \\nonumber \\cr
\n &= \\sqrt{a^2(d\\theta)^2\\cos^2\\theta+b^2(d\\theta)\\sin^2\\theta} \\nonumber \\cr
\n &= d\\theta\\sqrt{a^2\\cos^2\\theta+b^2\\sin^2\\theta} \\nonumber \\cr
\n &= d\\theta\\sqrt{a^2(1-\\sin^2\\theta)+b^2\\sin^2\\theta} \\nonumber \\cr
\n &= d\\theta\\sqrt{a^2+(b^2-a^2)\\sin^2\\theta} \\nonumber \\cr
\n &= d\\theta\\sqrt{a^2-(a^2-b^2)\\sin^2\\theta} \\nonumber \\\\
\n &= a\\,d\\theta \\sqrt{1-\\frac{a^2-b^2}{a^2}\\sin^2\\theta} \\label{eq:dssecond}
\n\\end{align}
\n\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n

(\\ref{eq:dssecond})\u5f0f\u306e\u6700\u5f8c\u3060\u3051\u3044\u307e\u3044\u3061\u30b7\u30f3\u30d7\u30eb\u306a\u5f0f\u3067\u306a\u3044\u611f\u3058\u3082\u3057\u307e\u3059\u304c\u3001$a$\u306f\u6955\u5186\u306e\u9577\u534a\u5f84\u3067\u3042\u308a\u3001$b$\u306f\u77ed\u534a\u5f84\u3067\u3042\u308b\u306e\u3067\u3001\u96e2\u5fc3\u7387$k$($e$\u3068\u66f8\u304f\u3053\u3068\u3082\u3042\u308a\u307e\u3059\u304c\u3001\u3053\u3053\u3067\u306f(\\ref{eq:secondellipticintegral})\u5f0f\u3068\u5408\u308f\u305b\u3066$k$\u3068\u66f8\u304d\u307e\u3059\u3002)\u3092
\n\\begin{align}
\n k = \\sqrt{\\frac{a^2-b^2}{a^2}} \\label{eq:eccentricity}
\n\\end{align}
\n\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u3088\u3063\u3066\u3001(\\ref{eq:eccentricity})\u5f0f\u3092(\\ref{eq:dssecond})\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\u2026
\n\\begin{align}
\n |PQ| &= a\\,d\\theta\\sqrt{1-k^2\\sin^2\\theta} \\label{eq:dsthird}
\n\\end{align}
\n\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u3053\u3053\u3067\u3001\u6955\u5186(\\ref{eq:ellipse})\u3068$y$\u8ef8\u306e\u6b63\u306e\u5411\u304d\u3068\u306e\u4ea4\u70b9\u3092$P_0$\u3001\u539f\u70b9$O$\u3092\u7aef\u70b9\u3068\u3059\u308b\u534a\u76f4\u7dda\u3092\u8003\u3048\u3001\u305d\u308c\u3068\u6955\u5186(\\ref{eq:ellipse})\u306e\u4ea4\u70b9\u3092$P_{\\varphi}$\u3068\u3057\u305f\u3068\u304d\u306b\u3001\u305d\u306e\u534a\u76f4\u7dda\u3068$y$\u8ef8\u306e\u6b63\u306e\u5411\u304d\u3068\u306e\u306a\u3059\u89d2\u3092$\\varphi$\u3068\u3057\u307e\u3059\u3002<\/p>\n

\u3053\u306e\u6642\u3001\u5f27$P_0P_{\\varphi}$\u306f(\\ref{eq:dsthird})\u5f0f\u3067\u6c42\u3081\u308b\u3053\u3068\u306e\u3067\u304d\u308b\u5fae\u5c0f\u306a\u7dda\u5206$|PQ|$\u3092\u3064\u306a\u304e\u5408\u308f\u305b\u305f\u3082\u306e\u306e\u7dcf\u548c\u3067\u3042\u308b\u3068\u8003\u3048\u308b\u3053\u3068\u304c\u3067\u304d\u308b\u306e\u3067\u3001\u5f27$P_0P_{\\varphi}$\u306e\u9577\u3055$|P_0P_{\\varphi}|$\u306f$|PQ|=ds$\u3068\u304a\u304f\u3068\u2026
\n\\begin{align}
\n |P_0P_{\\varphi}| &= \\int_{P_0P_{\\varphi}}ds \\nonumber \\cr
\n &= \\int_{0}^{\\varphi}a\\sqrt{1-k^2\\sin^2\\theta}\\,d\\theta \\nonumber \\cr
\n &= a\\int_{0}^{\\varphi}\\sqrt{1-k^2\\sin^2\\theta}\\,d\\theta \\nonumber \\\\
\n &= a\\,E(\\varphi, k) \\label{eq:dsfinal}
\n\\end{align}
\n\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

(\\ref{eq:dsfinal})\u5f0f\u3067$a=1$\u3068\u304a\u304f\u3068\u3001(\\ref{eq:secondellipticintegral})\u5f0f\u304c\u73fe\u308c\u307e\u3059\u3002$\\blacksquare$<\/p>\n

\u30cf\u30de\u308a\u304c\u3061\u306a\u5c0e\u51fa\u6cd5<\/h3>\n

\u3068\u3053\u308d\u304c\u3001\u5927\u5b66\u306e\u53d7\u9a13\u52c9\u5f37\u306e\u6642\u4ee3\u304b\u3089\u306e\u7656\u3067<\/p>\n

\u300c\u3048\u3063\u3068\u3001\u52d5\u5f84\u306f\u5de6\u56de\u308a\u304c\u6b63\u3060\u304b\u3089\u3001\u3063\u3068\u2026\u300d<\/p>\n

\u3068\u8003\u3048\u3066\u3001\u4e0b\u56f3\u306e\u3088\u3046\u306b\u70b9P\u306e\u5ea7\u6a19\u3092$(a\\cos\\theta,b\\sin\\theta)$\u3001\u70b9Q\u306e\u5ea7\u6a19\u3092$(a\\cos(\\theta+d\\theta),b\\sin(\\theta+d\\theta))$\u3068\u3068\u3063\u305f\u3068\u3057\u307e\u3059\u3002<\/p>\n

\"\"<\/a><\/p>\n

\u3059\u308b\u3068\u3001$|PQ|$\u306f\u2026
\n\\begin{align}
\n |PQ| &= \\sqrt{a^2\\{\\cos(\\theta+d\\theta)-\\cos\\theta\\}^2+b^2\\{\\sin(\\theta+d\\theta)-\\sin\\theta\\}^2} \\nonumber \\cr
\n &= \\sqrt{a^2\\{\\cos\\theta\\cos d\\theta-\\sin\\theta\\sin d\\theta-\\cos\\theta\\}^2+b^2\\{\\sin\\theta\\cos d\\theta+\\cos\\theta\\sin d\\theta-\\sin\\theta\\}^2} \\nonumber \\cr
\n &= \\sqrt{a^2\\{\\cos\\theta-d\\theta\\sin\\theta-\\cos\\theta\\}^2+b^2\\{\\sin\\theta+d\\theta\\cos\\theta-\\sin\\theta\\}^2} \\nonumber \\cr
\n &= \\sqrt{a^2\\{-d\\theta\\sin\\theta\\}^2+b^2\\{d\\theta\\cos\\theta\\}^2} \\nonumber \\cr
\n &= \\sqrt{a^2(d\\theta)^2\\sin^2\\theta+b^2(d\\theta)\\cos^2\\theta} \\nonumber \\cr
\n &= d\\theta\\sqrt{a^2\\sin^2\\theta+b^2\\cos^2\\theta} \\nonumber \\cr
\n &= d\\theta\\sqrt{a^2(1-\\cos^2\\theta)+b^2\\cos^2\\theta} \\nonumber \\cr
\n &= d\\theta\\sqrt{a^2+(b^2-a^2)\\cos^2\\theta} \\nonumber \\cr
\n &= d\\theta\\sqrt{a^2-(a^2-b^2)\\cos^2\\theta} \\nonumber \\cr
\n &= a\\,d\\theta\\sqrt{1-\\frac{a^2-b^2}{a^2}\\cos^2\\theta} \\nonumber \\\\
\n &= a\\,d\\theta\\sqrt{1-k^2\\cos^2\\theta} \\label{eq:dscosfirst}
\n\\end{align}
\n\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

(\\ref{eq:secondellipticintegral})\u5f0f\u3068\u306f\u5fae\u5999\u306b\u5f0f\u304c\u7570\u306a\u308a\u307e\u3059\u304c\u3001\u305d\u308c\u3082\u306e\u305d\u306e\u306f\u305a\u3001\u7a4d\u5206\u306e\u7d4c\u8def\u3092\u6955\u5186\u306e\u5f27\u4e0a\u306b\u304a\u3044\u3066\u9006\u5411\u304d\u306b\u3068\u3063\u3066\u3044\u308b\u3053\u3068\u306b\u8d77\u56e0\u3059\u308b\u76f8\u9055\u3067\u3059\u3002<\/p>\n

\u3068\u308a\u3042\u3048\u305a\u3053\u306e\u76f8\u9055\u306f\u7f6e\u3044\u3066\u304a\u3044\u3066\u3001$E(\\cdot)$\u3092\u7528\u3044\u305f\u5f62\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u306a\u3044\u304b\u8003\u3048\u3066\u307f\u307e\u3059\u3002<\/p>\n

(\\ref{eq:dsfinal})\u5f0f\u3092\u8a08\u7b97\u3057\u305f\u6642\u3068\u540c\u69d8\u306b$x$\u8ef8\u4e0a\u306b$P_0$\u3001\u6955\u5186\u306e\u5186\u5f27\u4e0a\u306b$P_{\\psi}$\u3092\u3068\u308b\u3053\u3068\u306b\u3059\u308b\u3068\u2026
\n\\begin{align}
\n |P_0P_{\\psi}| &= \\int_{P_0P_{\\varphi}}ds \\nonumber \\cr
\n &= \\int_{0}^{\\psi}a\\sqrt{1-k^2\\cos^2\\theta}\\,d\\theta \\nonumber \\\\
\n &= a\\int_{0}^{\\psi}\\sqrt{1-k^2\\cos^2\\theta}\\,d\\theta \\label{eq:dscosthird}
\n\\end{align}
\n\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u3053\u3053\u3067\u3001$\\psi \\in \\left[ 0, \\displaystyle\\frac{\\pi}{2} \\right]$\u3067\u3042\u308b\u3053\u3068\u306b\u7740\u76ee\u3057\u3001$t = \\displaystyle\\frac{\\pi}{2} – \\theta$\u3068\u304a\u3044\u3066(\\ref{eq:dscosthird})\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\u2026
\n\\begin{align}
\n a\\int_{0}^{\\psi}\\sqrt{1-k^2\\cos^2\\theta}\\,d\\theta &= -a\\int_{\\frac{\\pi}{2}}^{\\frac{\\pi}{2}-\\psi}\\sqrt{1-k^2\\sin^2t}\\,dt \\nonumber \\cr
\n &= a\\int_{\\frac{\\pi}{2}-\\psi}^{\\frac{\\pi}{2}}\\sqrt{1-k^2\\sin^2t}\\,dt \\nonumber \\cr
\n &= a\\left(\\int_{0}^{\\frac{\\pi}{2}}\\sqrt{1-k^2\\sin^2t}\\,dt\\,\\, – \\int_{0}^{\\frac{\\pi}{2}-\\psi}\\sqrt{1-k^2\\sin^2t}\\,dt\\right) \\nonumber \\\\
\n &= a\\left(E\\left(\\frac{\\pi}{2},k\\right)-E\\left(\\frac{\\pi}{2}-\\psi,k\\right)\\right) \\label{eq:dscosfinal}
\n\\end{align}
\n\u2026\u3068$E(\\cdot)$\u3092\u7528\u3044\u305f\u5f62\u3067\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002$\\blacksquare$<\/p>\n

\u307e\u3068\u3081<\/h2>\n

$E(\\cdot)$\u3092\u7528\u3044\u305f\u5f62\u3067\u8868\u3059\u3053\u3068\u306f\u3067\u304d\u308b\u306e\u3067\u3059\u304c\u3001\u3061\u3087\u3063\u3068\u56de\u308a\u304f\u3069\u304f\u3066\u3001\u304b\u3064\u76f4\u611f\u7684\u306b\u306f\u3044\u307e\u3044\u3061\u308f\u304b\u308a\u306b\u304f\u3044\u5f0f\u306b\u306a\u3063\u3066\u3057\u307e\u3046\u70b9\u306b\u3064\u3044\u3066\u306f\u3001\u7b2c2\u7a2e\u6955\u5186\u7a4d\u5206\u306e\u5f0f\u3092(\\ref{eq:secondellipticintegral})\u5f0f\u306e\u5f62\u5f0f($\\sin$\u304c\u73fe\u308c\u308b\u5f62\u5f0f)\u3068\u3057\u3066\u5b9a\u7fa9\u3057\u3066\u3057\u307e\u3063\u305f\u5148\u4eba\u306e\u4e2d\u306e\u4eba\u3092\u6068\u3080\u3088\u308a\u4ed6\u306f\u3042\u308a\u307e\u305b\u3093\u3002<\/p>\n

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