{"id":101113,"date":"2023-10-31T11:40:24","date_gmt":"2023-10-31T11:40:24","guid":{"rendered":"https:\/\/science-hub.click\/%E5%9B%9E%E8%BB%A2%E8%A1%8C%E5%88%97%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A9%B3%E3%81%97%E3%81%8F%E8%A7%A3%E8%AA%AC\/"},"modified":"2023-10-31T11:40:24","modified_gmt":"2023-10-31T11:40:24","slug":"%E5%9B%9E%E8%BB%A2%E8%A1%8C%E5%88%97%E3%81%AB%E3%81%A4%E3%81%84%E3%81%A6%E8%A9%B3%E3%81%97%E3%81%8F%E8%A7%A3%E8%AA%AC","status":"publish","type":"post","link":"https:\/\/science-hub.click\/?p=101113","title":{"rendered":"\u56de\u8ee2\u884c\u5217\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac"},"content":{"rendered":"

\u5c0e\u5165<\/h2>

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$$ {Q^t\\ Q = I = Q\\ Q^t} $$<\/div>\u305d\u3057\u3066
$$ {\\det Q = 1.~} $$<\/div><\/p>

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2 \u6b21\u5143\u304a\u3088\u3073 3 \u6b21\u5143\u3067\u306e\u56de\u8ee2<\/h2>

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\u4e8c\u6b21\u5143\u3067\u306f<\/span><\/h3>

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$$ { R(\\theta) = \\begin{bmatrix} \\cos \\theta & -\\sin \\theta \\\\[3pt] \\sin \\theta & \\cos \\theta \\\\ \\end{bmatrix} } $$<\/div> (\u89d2\u5ea6\u56de\u8ee2\u03b8)<\/dd><\/dl>

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\u6307\u5411\u6027\u306e\u3042\u308b\u5e73\u9762\u5185\u3067<\/b><\/p>

\n\"\"<\/figure>
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\n\"\"<\/figure>
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$$ { R(\\theta) = \\begin{bmatrix} \\cos \\theta & -\\sin \\theta \\\\[3pt] \\sin \\theta & \\cos \\theta \\\\ \\end{bmatrix} } $$<\/div> \uff08\u89d2\u5ea6\u03b8\u306e\u53cd\u6642\u8a08\u56de\u308a\uff09 <\/dd><\/dl>
$$ { R(-\\theta) = \\begin{bmatrix} \\cos \\theta & \\sin \\theta \\\\[3pt] -\\sin \\theta & \\cos \\theta \\\\ \\end{bmatrix} } $$<\/div> (\u89d2\u5ea6\u03b8\u306e\u6642\u8a08\u56de\u308a\u306e\u56de\u8ee2)<\/dd><\/dl>

\u901a\u5e38\u306e\u30ed\u30fc\u30c6\u30fc\u30b7\u30e7\u30f3<\/b><\/p>

90\u00b0 \u304a\u3088\u3073 180\u00b0 \u306e\u56de\u8ee2\u306b\u5bfe\u5fdc\u3059\u308b\u884c\u5217\u306f\u7279\u306b\u4fbf\u5229\u3067\u3059\u3002 <\/p>

$$ {\\begin{alignat}{1} R(90^\\circ) &= \\begin{bmatrix} 0 & -1 \\\\[3pt] 1 & 0 \\\\ \\end{bmatrix} \\end{alignat} } $$<\/div> (\u53cd\u6642\u8a08\u56de\u308a90\u00b0\u56de\u8ee2) <\/dd><\/dl>
$$ {\\begin{alignat}{1} R(180^\\circ) &= \\begin{bmatrix} -1 & 0 \\\\[3pt] 0 & -1 \\\\ \\end{bmatrix} \\end{alignat} } $$<\/div> (180\u00b0\u56de\u8ee2) <\/dd><\/dl>
$$ {\\begin{alignat}{1} R(270^\\circ) &= \\begin{bmatrix} 0 & 1 \\\\[3pt] -1 & 0 \\\\ \\end{bmatrix} \\end{alignat} } $$<\/div> (\u6642\u8a08\u56de\u308a\u306b90\u00b0\u56de\u8ee2)<\/dd><\/dl>

\u4e09\u6b21\u5143\u3067\u306f<\/span><\/h3>

\u57fa\u672c\u7684\u306a\u884c\u5217<\/span><\/h4>

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$$ {R_{{x}}(\\theta) = \\begin{bmatrix}1 & 0 & 0 \\\\ 0 & \\cos \\theta & -\\sin \\theta \\\\ 0 & \\sin \\theta & \\cos \\theta\\end{bmatrix}} $$<\/div><\/dd>
$$ {R_{{y}}(\\theta) = \\begin{bmatrix}\\cos \\theta & 0 & \\sin \\theta \\\\ 0 & 1 & 0 \\\\ -\\sin \\theta & 0 & \\cos \\theta\\end{bmatrix}} $$<\/div><\/dd>
<\/dd><\/dl>

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\u7a7a\u9593\u5fd7\u5411\u3067<\/span><\/h4>

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\u4e00\u822c\u7684\u306a\u5834\u5408\u306e\u56de\u8ee2\u884c\u5217<\/span><\/h4>

\u4ed6\u306e\u56de\u8ee2\u884c\u5217\u306f\u3001\u884c\u5217\u306e\u4e57\u7b97<\/a><\/span>\u3092\u4f7f\u7528\u3057\u3066\u57fa\u672c\u56de\u8ee2\u304b\u3089\u53d6\u5f97\u3055\u308c\u307e\u3059\u3002\u305f\u3068\u3048\u3070\u3001\u88fd\u54c1<\/p>

$$ {R_x(\\gamma) \\, R_y(\\beta) \\, R_z(\\alpha)\\,\\!} $$<\/div><\/dd><\/dl>

\u306f\u3001\u30e8\u30fc\u3001\u30d4\u30c3\u30c1\u3001\u30ed\u30fc\u30eb\u304c\u305d\u308c\u305e\u308c\u03b1<\/i> \u3001 \u03b2<\/i> \u3001 \u03b3<\/i>\u3067\u3042\u308b\u56de\u8ee2\u3092\u8868\u3057\u307e\u3059\u3002\u88fd\u54c1\u3082\u540c\u69d8\u306b\u3001 <\/p>

$$ {R_z(\\gamma) \\, R_x(\\beta) \\, R_z(\\alpha)\\,\\!} $$<\/div><\/dd><\/dl>

\u306f\u3001\u30aa\u30a4\u30e9\u30fc\u89d2\u304c\u03b1<\/i> \u3001 \u03b2<\/i> \u3001 \u03b3<\/i>\u3067\u3042\u308b\u56de\u8ee2\u3092\u8868\u3057\u307e\u3059 (\u30aa\u30a4\u30e9\u30fc\u89d2\u306ez<\/b> – x<\/b> – z<\/b>\u898f\u5247\u3092\u4f7f\u7528)\u3002<\/p>

\u56de\u8ee2\u8ef8<\/span><\/h4>

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\n\"\"<\/figure>
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$$ {\\overrightarrow u} $$<\/div>\u306f 3 \u3064\u306e\u6e96\u540c\u578b\u6027P<\/b> \u3001 (I – P)<\/b>\u304a\u3088\u3073Q \u3092<\/b>\u4f7f\u7528\u3057\u3066\u5206\u89e3\u3067\u304d\u307e\u3059 (\u753b\u50cf\u3092\u30af\u30ea\u30c3\u30af\u3059\u308b\u3068\u62e1\u5927\u3057\u307e\u3059)\u3002<\/div><\/div><\/div>

\u56de\u8ee2\u884c\u5217\u304b\u3089\u306e\u8ef8<\/b><\/p>

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$$ {R\\textbf{u} = \\textbf{u}} $$<\/div><\/dd><\/dl>

\u8ef8\u3068\u89d2\u5ea6\u304b\u3089\u306e\u56de\u8ee2\u884c\u5217<\/b><\/p>

\u5358\u4f4d\u30d9\u30af\u30c8\u30eb\u306b\u3088\u3063\u3066\u65b9\u5411\u4ed8\u3051\u3089\u308c\u305f\u8ef8\u306e\u5468\u308a\u306e\u56de\u8ee2\u884c\u5217R \u3092<\/i>\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002

$$ {\\overrightarrow u\\begin{pmatrix}{u_x}\\\\u_y\\\\u_z\\end{pmatrix}} $$<\/div> \uff08\u305d\u308c\u3067
$$ {u_x^{2} + u_y^{2} + u_z^{2} = 1} $$<\/div> ) \u3068\u89d2\u5ea6\u03b8<\/i> \u3002\u5f0f\u306f\u6b21\u306e\u3068\u304a\u308a\u3067\u3059\u3002 <\/p>
$$ {R = \\begin{bmatrix} u_x^2+(1-u_x^2)c & u_x u_y(1-c)-u_zs & u_x u_z(1-c)+u_ys \\\\[3pt] u_x u_y(1-c)+u_zs & u_y^2+(1-u_y^2)c & u_y u_z(1-c)-u_xs \\\\[3pt] u_x u_z(1-c)-u_ys & u_y u_z(1-c)+u_xs & u_z^2+(1-u_z^2)c \\end{bmatrix}} $$<\/div><\/dd><\/dl>

\u307e\u305f\u306f<\/p>

$$ {c = \\cos\\theta, \\qquad s = \\sin\\theta} $$<\/div><\/dd><\/dl>

3 \u6b21\u5143\u7a7a\u9593\u304c\u5f93\u6765\u306e\u65b9\u6cd5\u3067\u65b9\u5411\u4ed8\u3051\u3089\u308c\u3066\u3044\u308b\u5834\u5408\u3001\u3053\u306e\u56de\u8ee2\u306f\u3001\u65b9\u5411\u30d9\u30af\u30c8\u30eb\u304c\u6b21\u306e\u3088\u3046\u306b\u914d\u7f6e\u3055\u308c\u305f\u89b3\u5bdf\u8005\u306b\u3068\u3063\u3066\u306f\u53cd\u6642\u8a08\u56de\u308a\u306b\u306a\u308a\u307e\u3059\u3002

$$ {\\overrightarrow u} $$<\/div>\u305d\u306e\u65b9\u5411\u3092\u6307\u3057\u307e\u3059 (\u53f3\u624b\u306e\u6cd5\u5247)\u3002<\/p>

\u8ef8\u3068\u89d2\u5ea6\u306e\u516c\u5f0f\u306e\u7c21\u7565\u5316\u3055\u308c\u305f\u5f62\u5f0f<\/b><\/p>

\u3053\u306e\u5f0f\u306f\u6b21\u306e\u3088\u3046\u306b\u7c21\u7565\u5316\u3067\u304d\u307e\u3059\u3002 <\/p>

$$ {R = P +(I-P)\\cos \\theta + Q\\,\\sin \\theta} $$<\/div><\/dd><\/dl>

\u307e\u305f\u306f<\/p>

$$ {P = \\begin{bmatrix} u_x^2 & u_x u_y & u_x u_z \\\\[3pt] u_x u_y & u_y^2 & u_y u_z \\\\[3pt] u_x u_z & u_y u_z & u_z^2 \\end{bmatrix} = \\textbf{u} \\, \\textbf{u}^t,\\qquad I = \\begin{bmatrix} 1 & 0 & 0 \\\\[3pt] 0 & 1 & 0 \\\\[3pt] 0 & 0 & 1 \\end{bmatrix},\\qquad Q = \\begin{bmatrix} 0 & -u_z & u_y \\\\[3pt] u_z & 0 & -u_x \\\\[3pt] -u_y & u_x & 0 \\end{bmatrix} } $$<\/div><\/dd><\/dl>

I \u306f<\/i>3 \u00d7 3 \u306e\u5358\u4f4d\u884c\u5217<\/a><\/span>\u3067\u3059\u3002 Q<\/i>\u884c\u5217\u306f \u306e\u53cd\u5bfe\u79f0\u8868\u73fe\u3067\u3059\u3002

$$ {\\overrightarrow u} $$<\/div> \u3002\u884c\u5217P \u306f<\/i>\u56de\u8ee2\u8ef8\u4e0a\u306e\u6295\u5f71<\/a><\/span>\u3067\u3042\u308a\u3001 I<\/i> \u2013 P \u306f<\/i>\u56de\u8ee2\u8ef8\u306b\u76f4\u4ea4\u3059\u308b\u5e73\u9762\u4e0a\u306e\u6295\u5f71\u3067\u3059\u3002
$$ {\\overrightarrow u} $$<\/div> \u3002<\/p>

\u9ad8\u6b21\u5143\u306e\u843d\u3068\u3057\u7a74<\/span><\/h3>

\u4e0a\u8a18\u306f\u4efb\u610f\u306e\u6b21\u5143 n \u306b\u4e00\u822c\u5316\u3067\u304d\u307e\u3059\u3002\u56de\u8ee2\u306e\u300c\u8ef8\u300d\u306f\u3001\u3059\u3079\u3066\u306e\u30d9\u30af\u30c8\u30eb\u304c\u56de\u8ee2\u306b\u3088\u3063\u3066\u56fa\u5b9a\u3055\u308c\u308b\u6b21\u5143 n-2 \u306e\u30d9\u30af\u30c8\u30eb\u90e8\u5206\u7a7a\u9593<\/a><\/span>\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u3088\u3046\u306a\u90e8\u5206\u7a7a\u9593 A \u3067\u306f\u3001\u5171\u901a\u306e\u300c\u8ef8\u300dA \u306e\u56de\u8ee2\u306f\u3001\u56de\u8ee2\u306b\u76f4\u4ea4\u3059\u308b\u5e73\u9762\u306e\u56de\u8ee2\u306b\u5bfe\u5fdc\u3057\u307e\u3059\u3002\u3053\u306e\u8ef8\uff08\u305d\u3057\u3066\u540c\u3058\u65b9\u6cd5\u3067\u69cb\u6210\u3055\u308c\u307e\u3059\uff09\u3002<\/p>

\u3057\u304b\u3057\u30013 \u3092\u8d85\u3048\u308b\u6b21\u5143\u3067\u306f\u3001\u56de\u8ee2\u304c\u5fc5\u305a\u3057\u3082\u3053\u306e\u5f62\u5f0f\u306b\u306a\u308b\u308f\u3051\u3067\u306f\u306a\u3044\u3068\u3044\u3046\u65b0\u3057\u3044\u4e8b\u5b9f\u304c\u3042\u308a\u307e\u3059 (\u3064\u307e\u308a\u3001\u305d\u306e\u56fa\u5b9a\u30d9\u30af\u30c8\u30eb\u306e\u90e8\u5206\u7a7a\u9593\u306f\u3001\u53b3\u5bc6\u306b n-2 \u3088\u308a\u3082\u5c0f\u3055\u3044\u6b21\u5143\u306b\u306a\u308b\u53ef\u80fd\u6027\u304c\u5341\u5206\u306b\u3042\u308a\u307e\u3059)\u3002\u305d\u308c\u306f\u3001\u56de\u8ee2\u306e\u7a4d\u306b\u3059\u304e\u307e\u305b\u3093\u3002\u3053\u306e\u30d5\u30a9\u30fc\u30e0 (\u4ee5\u4e0b\u306e\u4f8b\u3092\u53c2\u7167)\u3002<\/p><\/div>

\u53c2\u8003\u8cc7\u6599<\/h2>
  1. \u0645\u0635\u0641\u0648\u0641\u0629 \u062f\u0648\u0631\u0627\u0646 \u2013 arabe<\/a><\/li>
  2. \u041c\u0430\u0442\u0440\u0438\u0446\u0430 \u043d\u0430 \u0440\u043e\u0442\u0430\u0446\u0438\u044f \u2013 bulgare<\/a><\/li>
  3. Matriu de rotaci\u00f3 \u2013 catalan<\/a><\/li>
  4. Drehmatrix \u2013 allemand<\/a><\/li>
  5. \u03a0\u03af\u03bd\u03b1\u03ba\u03b1\u03c2 \u03c0\u03b5\u03c1\u03b9\u03c3\u03c4\u03c1\u03bf\u03c6\u03ae\u03c2 \u2013 grec<\/a><\/li>
  6. Rotation matrix \u2013 anglais<\/a><\/li><\/ol><\/div>\n
    \n

    \n \u56de\u8ee2\u884c\u5217\u306b\u3064\u3044\u3066\u8a73\u3057\u304f\u89e3\u8aac\u30fb\u95a2\u9023\u52d5\u753b\n <\/h2>\n
    \n \n
    \n
    \n