{"id":2282,"date":"2018-06-26T23:37:06","date_gmt":"2018-06-26T14:37:06","guid":{"rendered":"https:\/\/pandanote.info\/?p=2282"},"modified":"2022-08-07T00:31:45","modified_gmt":"2022-08-06T15:31:45","slug":"%e5%a4%8f%e3%81%8c%e6%9d%a5%e3%82%8c%e3%81%b0%e6%80%9d%e3%81%84%e5%87%ba%e3%81%99%e3%80%81euler%e8%a7%92%e3%80%81%e5%9b%9b%e5%85%83%e6%95%b0%e3%80%82","status":"publish","type":"post","link":"https:\/\/pandanote.info\/?p=2282","title":{"rendered":"\u590f\u304c\u6765\u308c\u3070\u601d\u3044\u51fa\u3059\u3001Euler\u89d2\u3001\u56db\u5143\u6570(1): \u3061\u3087\u3063\u3068\u8a08\u7b97\u3057\u3066\u307f\u305f\u3002"},"content":{"rendered":"

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

\u30bf\u30a4\u30c8\u30eb\u306b\u66f8\u3044\u3066\u304a\u3044\u3066\u3044\u304d\u306a\u308a\u3053\u3093\u306a\u3053\u3068\u3092\u66f8\u304f\u306e\u3082\u306a\u3093\u3067\u3059\u304c\u3001Euler\u89d2\u306f\u5225\u306e\u6a5f\u4f1a\u306b\u53d6\u308a\u4e0a\u3052\u308b(\u30b8\u30f3\u30d0\u30eb\u30ed\u30c3\u30af\u3068\u304b\u3042\u307e\u308a\u8003\u3048\u305f\u304f\u306a\u3044\u306e\u3067\u2026 (\u00b4\u30fb\u03c9\u30fb\uff40))\u304b\u3082\u3057\u308c\u306a\u3044\u3053\u3068\u306b\u3057\u3066\u3001\u3053\u306e\u8a18\u4e8b\u3067\u306f\u56db\u5143\u6570\u306b\u3064\u3044\u3066\u601d\u3044\u51fa\u3057\u305f\u3053\u3068\u3092\u66f8\u3044\u3066\u3044\u304d\u307e\u3059\u3002<\/p>\n

\u56db\u5143\u6570\u3068\u306f?<\/h2>\n

\u3068\u308a\u3042\u3048\u305a\u3001\u8aac\u660e\u3002<\/h3>\n

\u6570\u5b66\u7684\u306b\u53b3\u5bc6\u306a\u8aac\u660e\u3067\u306f\u306a\u3044\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u304c\u3001\u8aac\u660e\u3057\u3088\u3046 \ud83d\ude42<\/p>\n

\u56db\u5143\u6570\u3068\u306f\u7c21\u5358\u306b\u8a00\u3046\u3068\u8907\u7d20\u6570\u3092\u62e1\u5f35\u3057\u305f\u6570\u4f53\u7cfb\u3067\u3042\u308a\u3001\u8907\u7d20\u6570\u3067\u306f1\u500b\u3067\u3042\u3063\u305f\u865a\u6570\u5358\u4f4d\u304c\u56db\u5143\u6570\u3067\u306f3\u500b\u306b\u306a\u3063\u3066\u3044\u307e\u3059\u3002<\/p>\n

\u4e00\u822c\u306b\u56db\u5143\u6570$x$\u306f$p,q,r,s \\in \\mathbb{R}$\u3092\u7528\u3044\u3066$x = p+qi+rj+sk$\u3068\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u56db\u5143\u6570\u306e3\u500b\u306e\u865a\u6570\u5358\u4f4d(1\u4ee5\u5916\u306e\u57fa\u5e95\u5143)\u3092$i,j,k$\u3068\u7f6e\u304f\u3068\u3001\u4ee5\u4e0b\u306e\u95a2\u4fc2\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n

\\begin{eqnarray}
\ni^2 &=& j^2 = k^2 = ijk = -1 \\label{eq:quarterniondef}
\n\\end{eqnarray}<\/p>\n

\u3063\u3066\u3001\u30cf\u30df\u30eb\u30c8\u30f3\u304c1843\u5e74\u306b\u6700\u521d\u306b\u8003\u3048\u305f\u5f0f\u305d\u306e\u307e\u307e\u3067\u3059\u304c(\u6c57)\u3001\u3068\u306b\u304b\u304f\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n

\u305d\u308c\u3067\u3001\u4f55\u306b\u4f7f\u3048\u308b\u306e?<\/h3>\n

\u7269\u4f53\u306e\u59ff\u52e2\u3092Euler\u89d2\u3092\u4f7f\u3063\u3066\u8868\u73fe\u3059\u308b\u3068\u3001\u59ff\u52e2\u306b\u5bfe\u3059\u308b\u89d2\u5ea6\u304c\u6c7a\u307e\u3089\u306a\u3044\u7279\u7570\u70b9\u304c\u5fc5\u305a\u5b58\u5728\u3057\u307e\u3059\u304c\u3001\u56db\u5143\u6570\u3067\u59ff\u52e2\u3092\u8868\u3059\u3068\u3001\u305d\u306e\u3088\u3046\u306a\u554f\u984c\u3092\u56de\u907f\u3059\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u305d\u3053\u3067\u3001\u4eba\u5de5\u885b\u661f\u3084\u5b87\u5b99\u63a2\u67fb\u6a5f\u3001\u3055\u3089\u306b\u306f3\u6b21\u5143CG\u306b\u304a\u3051\u308b\u7269\u4f53\u306e\u904b\u52d5\u306a\u3069\u3068\u3044\u3063\u305f\u3001\u3069\u3046\u3044\u3046\u59ff\u52e2\u3092\u53d6\u308b\u304b\u304c\u308f\u304b\u3089\u306a\u304b\u3063\u305f\u308a\u3001\u59ff\u52e2\u306e\u5909\u5316\u304c\u5fae\u5c0f\u3067\u3042\u308b\u3068\u306f\u9650\u3089\u306a\u3044\u7269\u4f53\u306e\u59ff\u52e2\u306e\u8868\u73fe\u3084\u5236\u5fa1\u306a\u3069\u306b\u4f7f\u3048\u307e\u3059\u3002<\/p>\n

\u53d6\u6271\u4e0a\u306e\u6ce8\u610f<\/h3>\n

\u56db\u5143\u6570\u3092\u6271\u3046\u4e0a\u3067\u6700\u3082\u6ce8\u610f\u3057\u306a\u3051\u308c\u3070\u306a\u3089\u306a\u3044\u306e\u306f\u3001<\/p>\n

\u300c\u4e57\u6cd5\u306b\u95a2\u3059\u308b\u4ea4\u63db\u5247\u304c\u6210\u7acb\u3057\u306a\u3044\u3002\u300d<\/strong><\/p>\n

\u3053\u3068\u3067\u3057\u3087\u3046\u3002\u884c\u5217\u306e\u6f14\u7b97\u307f\u305f\u3044\u3067\u3059\u306d\u3002<\/p>\n

\u305d\u3053\u3067\u3001(\\ref{eq:quarterniondef})\u5f0f\u306e\u95a2\u4fc2\u5f0f\u3092\u4f7f\u3063\u3066\u4e57\u6cd5\u306b\u95a2\u3059\u308b\u4ea4\u63db\u5247\u304c\u6210\u308a\u7acb\u3064\u304b\u3069\u3046\u304b\u8abf\u3079\u3066\u307f\u307e\u3059\u3002<\/p>\n

(\\ref{eq:quarterniondef})\u5f0f\u306e\u7b49\u5f0f\u306e\u3046\u3061\u3001$ijk = -1$\u306e\u4e21\u8fba\u306b\u53f3\u304b\u3089$-k$\u3092\u304b\u3051\u308b\u3068\u3001(\\ref{eq:ijeqk})\u5f0f\u304c\u6c42\u307e\u308a\u307e\u3059\u3002<\/p>\n

\\begin{eqnarray}
\nij &=& k \\label{eq:ijeqk}
\n\\end{eqnarray}<\/p>\n

\u307e\u305f\u3001$ijk = -1$\u306e\u4e21\u8fba\u306b\u5de6\u304b\u3089$-i$\u3092\u304b\u3051\u308b\u3068<\/p>\n

\\begin{eqnarray}
\njk &=& i \\label{eq:jkeqi}
\n\\end{eqnarray}<\/p>\n

\u306b\u306a\u3063\u305f\u308a\u3057\u307e\u3059\u3002\u3055\u3089\u306b(\\ref{eq:jkeqi})\u5f0f\u306e\u4e21\u8fba\u306e\u5de6\u304b\u3089$j$\u3092\u304b\u3051\u308b\u3068\u3001(\\ref{eq:keqji})\u5f0f\u304c\u5f97\u3089\u308c\u307e\u3059\u3002<\/p>\n

\\begin{eqnarray}
\n-k &=& ji \\label{eq:keqji}
\n\\end{eqnarray}<\/p>\n

(\\ref{eq:ijeqk})\u5f0f\u53ca\u3073(\\ref{eq:keqji})\u5f0f\u3092\u6bd4\u8f03\u3059\u308b\u3068\u3001$i,j$\u306e\u9806\u5e8f\u3092\u5165\u308c\u66ff\u3048\u308b\u3068\u8a08\u7b97\u7d50\u679c\u306e\u7b26\u53f7\u304c\u5165\u308c\u66ff\u308f\u308b\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002\u3088\u3063\u3066\u3001$i,j$\u306b\u3064\u3044\u3066\u306f\u4e57\u6cd5\u306b\u95a2\u3059\u308b\u4ea4\u63db\u5247\u304c\u6210\u308a\u7acb\u305f\u306a\u3044\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

\u307e\u305f\u3001(\\ref{eq:ijeqk})\u5f0f\u306e\u4e21\u8fba\u306b\u5de6\u304b\u3089$j$\u3092\u304b\u3051\u308b\u3068\u3001<\/p>\n

\\begin{eqnarray}
\n-i &=& kj \\label{eq:ieqkj}
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308b\u306e\u3067\u3001(\\ref{eq:jkeqi})\u5f0f\u53ca\u3073(\\ref{eq:ieqkj})\u5f0f\u3088\u308a$j,k$\u306b\u3064\u3044\u3066\u3082\u4e57\u6cd5\u306b\u95a2\u3059\u308b\u4ea4\u63db\u5247\u304c\u6210\u308a\u7acb\u305f\u306a\u3044\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

$k,i$\u306e\u5834\u5408\u306b\u3064\u3044\u3066\u3082(\\ref{eq:keqji})\u5f0f\u306e\u4e21\u8fba\u306b$-i$\u3092\u5de6\u304b\u3089\u304b\u3051\u308b\u3068<\/p>\n

\\begin{eqnarray}
\nki &=& j \\label{eq:kieqj}
\n\\end{eqnarray}<\/p>\n

\u304c\u3001(\\ref{eq:jkeqi})\u5f0f\u306e\u4e21\u8fba\u306b\u53f3\u304b\u3089$k$\u3092\u304b\u3051\u308b\u3053\u3068\u3067\u3001<\/p>\n

\\begin{eqnarray}
\n-j &=& ik \\label{eq:jeqik}
\n\\end{eqnarray}<\/p>\n

\u304c\u305d\u308c\u305e\u308c\u5f97\u3089\u308c\u308b\u306e\u3067\u3001(\\ref{eq:kieqj})\u5f0f\u53ca\u3073(\\ref{eq:jeqik})\u5f0f\u3088\u308a\u4e57\u6cd5\u306b\u95a2\u3059\u308b\u4ea4\u63db\u5247\u304c\u6210\u308a\u7acb\u305f\u306a\u3044\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

\u3057\u305f\u304c\u3063\u3066\u30011\u4ee5\u5916\u306e\u57fa\u5e95\u5143$i,j,k$\u306b\u3064\u3044\u3066\u306f\u4e57\u6cd5\u306b\u95a2\u3059\u308b\u4ea4\u63db\u5247\u304c\u6210\u308a\u7acb\u305f\u306a\u3044\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

(\\ref{eq:quarterniondef})\u5f0f\u304b\u3089(\\ref{eq:jeqik})\u5f0f\u307e\u3067\u306e\u7d50\u679c\u53ca\u30731\u3068$i,j,k$\u306e\u7a4d\u306f\u4ee5\u4e0b\u306e\u8868\u306e\u3088\u3046\u306b\u307e\u3068\u3081\u308b\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002<\/p>\n\n\n\n\n\n\n\n
<\/th>\n$1$<\/th>\n$i$<\/th>\n$j$<\/th>\n$k$<\/th>\n<\/tr>\n
$1$<\/th>\n$1$<\/td>\n$i$<\/td>\n$j$<\/td>\n$k$<\/td>\n<\/tr>\n
$i$<\/th>\n$i$<\/td>\n$-1$<\/td>\n$k$<\/td>\n$-j$<\/td>\n<\/tr>\n
$j$<\/th>\n$j$<\/td>\n$-k$<\/td>\n$-1$<\/td>\n$i$<\/td>\n<\/tr>\n
$k$<\/th>\n$k$<\/td>\n$j$<\/td>\n$-i$<\/td>\n$-1$<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n

\u4e57\u7b97\u306e\u6f14\u7b97\u3002<\/h2>\n

\u3053\u3053\u307e\u3067\u306e\u8aac\u660e\u3067\u56db\u5143\u6570\u306e\u771f\u9ac4\u3092\u7406\u89e3\u3057\u3066\u3044\u305f\u3060\u3051\u305f\u3068\u601d\u3044\u307e\u3059\u306e\u3067\u3001\u56db\u5143\u6570\u3069\u3046\u3057\u306e\u4e57\u7b97\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3059\u3002
\n\u56db\u5143\u6570$q_1,q_2$\u306e\u4e57\u7b97\u306f\u3001<\/p>\n

\\begin{eqnarray}
\nq_1&=&a_1+b_1i+c_1j+d_1k \\nonumber \\\\
\nq_2&=&a_2+b_2i+c_2j+d_2k \\nonumber
\n\\end{eqnarray}<\/p>\n

\u3068\u304a\u304f\u3068\u3001<\/p>\n

\\begin{eqnarray}
\nq_1q_2 &=& a_1a_2-b_1b_2-c_1c_2-d_1d_2 \\nonumber \\\\
\n&{}& +a_1b_2i+a_1c_2j+a_1d_2k \\nonumber \\\\
\n&{}& +b_1a_2i+b_1c_2k-b_1d_2j \\nonumber \\\\
\n&{}& +c_1a_2j-c_1b_2k+c_1d_2i \\nonumber \\\\
\n&{}& +d_1a_2k+d_1b_2j-d_1c_2i \\nonumber \\\\
\n&=& a_1a_2-b_1b_2-c_1c_2-d_1d_2 \\nonumber \\\\
\n&{}& +(a_1b_2+b_1a_2+c_1d_2-d_1c_2)i \\nonumber \\\\
\n&{}& +(a_1c_2-b_1d_2+c_1a_2+d_1b_2)j \\nonumber \\\\
\n&{}& +(a_1d_2+b_1c_2-c_1b_2+d_1a_2)k \\label{eq:qmul}
\n\\end{eqnarray}<\/p>\n

\u3068\u8a08\u7b97\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u2026\u3068\u3044\u3063\u3066\u3082\u3001(\\ref{eq:qmul})\u5f0f\u306e\u307e\u307e\u3060\u3068$i,j,k$\u306b\u3064\u3044\u3066\u6574\u7406\u3057\u305f\u306b\u3082\u304b\u304b\u308f\u3089\u305a\u3001\u6587\u5b57\u6570\u304c\u591a\u3059\u304e\u3066\u7e41\u96d1\u306a\u611f\u3058\u304c\u3057\u307e\u3059\u3002<\/p>\n

\u305d\u3053\u3067\u3001\u5fdc\u7528\u9762\u3082\u898b\u8d8a\u3057\u3066\u3082\u3046\u5c11\u3057\u898b\u901a\u3057\u306e\u826f\u3055\u305d\u3046\u306a\u66f8\u304d\u65b9\u304c\u3067\u304d\u306a\u3044\u304b\u8003\u3048\u3066\u307f\u307e\u3059\u3002$\\boldsymbol{b}_1 = (b_1,c_1,d_1), \\boldsymbol{i}_1 = \\displaystyle\\left(
\n \\begin{array}{c}
\n i_1 \\\\
\n j_1 \\\\
\n k_1
\n \\end{array}
\n \\right)$\u3068\u304a\u304f\u3068\u3001$q_1 = a_1+\\boldsymbol{b}_1\\cdot\\boldsymbol{i}_1$\u3068\u66f8\u3051\u307e\u3059\u3002$q_2$\u306b\u3064\u3044\u3066\u3082\u540c\u69d8\u306b$q_2 = a_2+\\boldsymbol{b}_2\\cdot\\boldsymbol{i}_2$\u3068\u66f8\u3051\u307e\u3059\u3002<\/p>\n

(\\ref{eq:qmul})\u5f0f\u306e\u53f3\u8fba\u3092$a_1, a_2$\u53ca\u3073$i,j,k$\u304c\u5165\u3089\u306a\u3044(=\u5b9f\u6570\u306b\u306a\u308b)\u9805\u306b\u7740\u76ee\u3057\u3066\u518d\u5ea6\u6574\u7406\u3057\u76f4\u3059\u3068\u3001<\/p>\n

\\begin{eqnarray}
\nq_1q_2 &=& a_1a_2-b_1b_2-c_1c_2-d_1d_2 \\nonumber \\\\
\n&{}& +a_1(b_2i+c_2j+d_2k) \\nonumber \\\\
\n&{}& +a_2(b_1i+c_1j+d_1k) \\nonumber \\\\
\n&{}& +(c_1d_2-d_1c_2)i \\nonumber \\\\
\n&{}& +(-b_1d_2+d_1b_2)j \\nonumber \\\\
\n&{}& +(b_1c_2-c_1b_2)k \\nonumber \\\\
\n&=& a_1a_2-\\boldsymbol{b}_1\\cdot\\boldsymbol{b}_2 +
\n a_1(\\boldsymbol{b}_2\\cdot\\boldsymbol{i}) + a_2(\\boldsymbol{b}_1\\cdot\\boldsymbol{i}) \\nonumber \\\\
\n&{}& +(c_1d_2-d_1c_2)i \\nonumber \\\\
\n&{}& +(-b_1d_2+d_1b_2)j \\nonumber \\\\
\n&{}& +(b_1c_2-c_1b_2)k \\nonumber \\\\
\n&=& a_1a_2-\\boldsymbol{b}_1\\cdot\\boldsymbol{b}_2 +
\n a_1(\\boldsymbol{b}_2\\cdot\\boldsymbol{i}) + a_2(\\boldsymbol{b}_1\\cdot\\boldsymbol{i}) + (\\boldsymbol{b}_1 \\times \\boldsymbol{b}_2) \\cdot \\boldsymbol{i} \\label{eq:qmulimproved}
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308a\u307e\u3059\u3002\u5185\u7a4d\u3084\u5916\u7a4d\u3092\u7528\u3044\u3066\u4f55\u3068\u304b\u8868\u73fe\u3067\u304d\u308b\u306e\u3067\u3001\u5c11\u3057\u308f\u304b\u308a\u3084\u3059\u3044\u5f0f\u306b\u306a\u3063\u305f\u304b\u3068\u601d\u3044\u307e\u3059\u3002\uff08\uff40\u30fb\u03c9\u30fb\u00b4\uff09<\/p>\n

(\\ref{eq:qmulimproved})\u5f0f\u306e\u53f3\u8fba\u7b2c3\u5f0f\u306e\u6700\u5f8c\u306e\u9805\u304c\u865a\u6570\u6210\u5206\u306e\u5916\u7a4d\u306e\u5f0f\u306b\u306a\u3063\u3066\u3044\u308b\u305f\u3081\u3001\u4e00\u822c\u306e\u56db\u5143\u6570$q_1,q_2$\u306b\u3064\u3044\u3066\u306f\u7a4d\u306e\u4ea4\u63db\u5247\u304c\u6210\u308a\u7acb\u305f\u306a\u3044\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

\u56db\u5143\u6570\u3092\u7528\u3044\u305f\u4e09\u6b21\u5143\u5ea7\u6a19\u7cfb\u306e\u56de\u8ee2\u5909\u63db\u306e\u8868\u73fe<\/h2>\n

\u524d\u9805\u3067\u4f55\u3068\u304b\u56db\u5143\u6570\u306e\u4e57\u7b97\u304c\u3067\u304d\u305f\u306e\u3067\u3001\u3053\u308c\u3092\u4f7f\u3063\u3066\u56db\u5143\u6570\u3092\u7528\u3044\u305f\u4e09\u6b21\u5143\u5ea7\u6a19\u7cfb\u306e\u56de\u8ee2\u5909\u63db\u3092\u884c\u3046\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n

\u3053\u3053\u3067\u7a81\u7136\u3067\u3059\u304c\u3001\u5358\u4f4d\u30d9\u30af\u30c8\u30eb$\\boldsymbol{u} = \\displaystyle\\left(
\n \\begin{array}{c}
\n u_x \\\\
\n u_y \\\\
\n u_z
\n \\end{array}
\n \\right), ||\\boldsymbol{u}|| = 1$($||\\cdots||$\u306f$\\cdots$\u306e\u30ce\u30eb\u30e0\u3092\u8868\u3057\u307e\u3059\u3002)\u306e\u5468\u308a\u3092\u53f3\u306d\u3058\u304c\u9032\u3080\u65b9\u5411\u306b$\\theta$\u3060\u3051\u56de\u8ee2\u3055\u305b\u308b\u56de\u8ee2\u79fb\u52d5\u3092\u8003\u3048\u3001\u3055\u3089\u306b\u3001$q = \\cos\\displaystyle\\frac{\\theta}{2} + \\boldsymbol {u}\\boldsymbol{i}\\sin\\displaystyle\\frac{\\theta}{2}$\u3068\u304a\u304d\u307e\u3059\u3002\u307e\u305f$q$\u306e\u5171\u5f79\u56db\u5143\u6570\u3092$\\overline{q} = \\cos\\displaystyle\\frac{\\theta}{2} – \\boldsymbol{u}\\boldsymbol{i}\\sin\\displaystyle\\frac{\\theta}{2}$\u3068\u3057\u307e\u3059\u3002<\/p>\n

\u3053\u3053\u3067\u3001\u3042\u308b\u56db\u5143\u6570$x = w + \\boldsymbol{r}\\boldsymbol{i}, w \\in \\mathbb{R}, \\boldsymbol{r} \\in \\mathbb{R}^3, \\boldsymbol{i} = \\displaystyle\\left(
\n \\begin{array}{c}
\n i \\\\
\n j \\\\
\n k
\n \\end{array}
\n \\right)$\u306b\u3064\u3044\u3066\u3001\u4ee5\u4e0b\u306e\u5f0f\u306b\u3088\u308a\u8a08\u7b97\u3067\u304d\u308b$x^{\\prime}$\u3092\u6c42\u3081\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002<\/p>\n

\\begin{eqnarray}
\nx^{\\prime} &=& qx\\overline{q} \\label{eq:qxq}
\n\\end{eqnarray}<\/p>\n

\u5c11\u3005\u8a08\u7b97\u304c\u9762\u5012\u3067\u306f\u3042\u308a\u307e\u3059\u304c\u3001(\\ref{eq:qmulimproved})\u5f0f\u306e\u95a2\u4fc2\u3092\u7528\u3044\u3066(\\ref{eq:qxq})\u5f0f\u306e\u53f3\u8fba\u3092\u9811\u5f35\u3063\u3066\u8a08\u7b97\u3059\u308b\u3068\u3001<\/p>\n

\\begin{eqnarray}
\n\\def\\coshalftheta{\\cos\\frac{\\theta}{2}}
\n\\def\\sinhalftheta{\\sin\\frac{\\theta}{2}}
\n\\def\\bmu{\\boldsymbol{u}}
\n\\def\\bmi{\\boldsymbol{i}}
\n\\def\\bmr{\\boldsymbol{r}}
\nx^{\\prime} &=& \\left[\\coshalftheta + \\bmu\\bmi\\sinhalftheta\\right]\\left[w +\\bmr\\bmi\\right]\\left[\\coshalftheta – \\bmu\\bmi\\sinhalftheta\\right] \\nonumber \\\\
\n&=& \\left[w\\coshalftheta – \\bmu\\cdot\\bmr\\sinhalftheta+\\left[\\bmr\\coshalftheta+w\\bmu\\sinhalftheta+(\\bmu\\times\\bmr)\\sinhalftheta\\right]\\bmi\\right] \\cdot \\nonumber \\\\
\n&{}& \\left[\\coshalftheta – \\bmu\\bmi\\sinhalftheta\\right] \\nonumber \\\\
\n&=& w\\cos^2\\frac{\\theta}{2}-\\bmu\\cdot\\bmr\\sinhalftheta\\coshalftheta \\nonumber \\\\
\n&{}& +\\left[\\bmr\\coshalftheta+w\\bmu\\sinhalftheta+(\\bmu\\times\\bmr)\\sinhalftheta\\right]\\cdot\\bmu\\sinhalftheta \\nonumber \\\\
\n&{}& -\\left[w\\coshalftheta-\\bmu\\cdot\\bmr\\sinhalftheta\\right]\\cdot\\bmu\\bmi\\sinhalftheta \\nonumber \\\\
\n&{}& +\\left[\\bmr\\coshalftheta+w\\bmu\\sinhalftheta+(\\bmu\\times\\bmr)\\sinhalftheta\\right]\\bmi\\coshalftheta \\nonumber \\\\
\n&{}& -\\left[\\left[\\bmr\\coshalftheta+w\\bmu\\sinhalftheta+(\\bmu\\times\\bmr)\\sinhalftheta\\right]\\times\\bmu\\right]\\bmi\\sinhalftheta \\label{eq:xprime}
\n\\end{eqnarray}<\/p>\n

(\\ref{eq:xprime})\u5f0f\u3092\u898b\u308b\u9650\u308a\u3001\u3053\u306e\u8a08\u7b97\u306e\u5148\u884c\u304d\u306b\u304b\u306a\u308a\u4e0d\u5b89\u3092\u899a\u3048\u308b\u611f\u3058\u3082\u3057\u307e\u3059\u304c\u3001\u307e\u305a\u5b9f\u90e8\u306e\u4fc2\u6570\u3092\u307e\u3068\u3081\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002$\\boldsymbol{u}\\times\\boldsymbol{r}$\u3068$\\boldsymbol{u}$\u306f\u76f4\u4ea4\u3059\u308b\u3053\u3068\u304b\u3089\u3001$(\\boldsymbol{u}\\times\\boldsymbol{r})\\cdot\\boldsymbol{u} = 0$\u3068\u306a\u308a\u307e\u3059\u306e\u3067\u3001<\/p>\n

\\begin{eqnarray}
\n\\def\\coshalftheta{\\cos\\frac{\\theta}{2}}
\n\\def\\sinhalftheta{\\sin\\frac{\\theta}{2}}
\n\\def\\bmu{\\boldsymbol{u}}
\n\\def\\bmi{\\boldsymbol{i}}
\n\\def\\bmr{\\boldsymbol{r}}
\n&{}&w\\cos^2\\frac{\\theta}{2}-\\bmu\\cdot\\bmr\\sinhalftheta\\coshalftheta \\nonumber \\\\
\n&{}& +\\left[\\bmr\\coshalftheta+w\\bmu\\sinhalftheta+(\\bmu\\times\\bmr)\\sinhalftheta\\right]\\cdot\\bmu\\sinhalftheta \\nonumber \\\\
\n&=& w\\cos^2\\frac{\\theta}{2}-\\bmu\\cdot\\bmr\\sinhalftheta\\coshalftheta+\\bmr\\cdot\\bmu\\coshalftheta\\sinhalftheta+w\\sin^2\\frac{\\theta}{2} \\nonumber \\\\
\n&=& w \\label{eq:w}
\n\\end{eqnarray}<\/p>\n

\u3068\u306a\u308b\u306e\u3067\u3001\u5b9f\u90e8\u306e\u5024($w$)\u306f(\\ref{eq:qxq})\u5f0f\u306e\u8a08\u7b97\u3067\u306f\u5909\u5316\u3057\u306a\u3044\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

(\\ref{eq:w})\u5f0f\u3067\u5b9f\u90e8\u306f\u8a08\u7b97\u3067\u304d\u305f\u306e\u3067\u3001\u6b21\u306f\u865a\u90e8\u3092\u307e\u3068\u3081\u308b\u3053\u3068\u3092\u8003\u3048\u307e\u3059\u3002\u865a\u90e8\u306e\u9805\u306e\u3046\u3061$w$\u304c\u73fe\u308c\u308b\u9805\u306f3\u9805\u3042\u308a\u307e\u3059\u304c\u3001\u3053\u308c\u3089\u306e\u548c\u306f$0$\u3068\u306a\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u6b8b\u308a\u306e\u9805\u3092\u5b9f\u90e8\u307e\u3067\u542b\u3081\u3066\u8868\u3059\u3068\u3001(\\ref{eq:im})\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n

\\begin{eqnarray}
\n\\def\\coshalftheta{\\cos\\frac{\\theta}{2}}
\n\\def\\sinhalftheta{\\sin\\frac{\\theta}{2}}
\n\\def\\bmu{\\boldsymbol{u}}
\n\\def\\bmi{\\boldsymbol{i}}
\n\\def\\bmr{\\boldsymbol{r}}
\nx^{\\prime} &=& w+\\left[(\\bmu\\cdot\\bmr)\\cdot\\bmu\\sin^2\\frac{\\theta}{2}+\\bmr\\cos^2\\frac{\\theta}{2}+(\\bmu\\times\\bmr)\\sinhalftheta\\coshalftheta\\right. \\nonumber \\\\
\n&{}& \\left.-(\\bmr\\times\\bmu)\\coshalftheta\\sinhalftheta-\\left[(\\bmu\\times\\bmr)\\times\\bmu\\right]\\sin^2\\frac{\\theta}{2}\\right]\\bmi \\nonumber \\\\
\n&=& w+\\left[(\\bmu\\times\\bmr)\\sin\\theta+\\bmr\\cos^2\\frac{\\theta}{2}+(\\bmu\\cdot\\bmr)\\cdot\\bmu\\sin^2\\frac{\\theta}{2}\\right. \\nonumber \\\\
\n&{}& \\left.-\\left[(\\bmu\\times\\bmr)\\times\\bmu\\right]\\sin^2\\frac{\\theta}{2}\\right]\\bmi\\label{eq:im}
\n\\end{eqnarray}<\/p>\n

\u3053\u3053\u3067\u3001$\\boldsymbol{r}$\u3092$\\boldsymbol{u}$\u306b\u5e73\u884c\u306a\u6210\u5206$\\boldsymbol{r}_u$\u3068$\\boldsymbol{u}$\u3068\u76f4\u4ea4\u3059\u308b\u6210\u5206$\\boldsymbol{r}_l$\u306b\u5206\u89e3\u3057\u307e\u3059\u3002\u3059\u308b\u3068\u3001<\/p>\n

\\begin{eqnarray}
\n\\def\\coshalftheta{\\cos\\frac{\\theta}{2}}
\n\\def\\sinhalftheta{\\sin\\frac{\\theta}{2}}
\n\\def\\bmu{\\boldsymbol{u}}
\n\\def\\bmi{\\boldsymbol{i}}
\n\\def\\bmr{\\boldsymbol{r}}
\n\\bmr &=& \\bmr_u+\\bmr_l \\label{eq:decomposition}
\n\\end{eqnarray}<\/p>\n

\u3068\u66f8\u3051\u308b\u306e\u3067\u3001(\\ref{eq:bmru})\u53ca\u3073(\\ref{eq:bmrl})\u5f0f\u304c\u6210\u308a\u7acb\u3061\u307e\u3059\u3002<\/p>\n

\\begin{eqnarray}
\n\\def\\coshalftheta{\\cos\\frac{\\theta}{2}}
\n\\def\\sinhalftheta{\\sin\\frac{\\theta}{2}}
\n\\def\\bmu{\\boldsymbol{u}}
\n\\def\\bmi{\\boldsymbol{i}}
\n\\def\\bmr{\\boldsymbol{r}}
\n(\\bmu\\cdot\\bmr)\\bmu &=& \\left[\\bmu\\cdot(\\bmr_u+\\bmr_l)\\right]\\cdot\\bmu \\nonumber \\\\
\n&=& || \\bmr_u || \\bmu \\nonumber \\\\
\n&=& \\bmr_u \\label{eq:bmru} \\\\
\n(\\bmu\\times\\bmr)\\times\\bmu &=& (\\bmu\\times\\bmr_l)\\times\\bmu \\nonumber \\\\
\n&=& \\bmr_l \\label{eq:bmrl}
\n\\end{eqnarray}<\/p>\n

(\\ref{eq:decomposition})\u5f0f(\\ref{eq:bmru})\u53ca\u3073(\\ref{eq:bmrl})\u306e\u5404\u5f0f\u3092(\\ref{eq:im})\u5f0f\u306b\u4ee3\u5165\u3059\u308b\u3068\u3001<\/p>\n

\\begin{eqnarray}
\n\\def\\coshalftheta{\\cos\\frac{\\theta}{2}}
\n\\def\\sinhalftheta{\\sin\\frac{\\theta}{2}}
\n\\def\\bmu{\\boldsymbol{u}}
\n\\def\\bmi{\\boldsymbol{i}}
\n\\def\\bmr{\\boldsymbol{r}}
\nx^{\\prime} &=& w+\\left[(\\bmu\\times\\bmr_l)\\sin\\theta+(\\bmr_u+\\bmr_l)\\cos^2\\frac{\\theta}{2}+\\bmr_u\\sin^2\\frac{\\theta}{2}\\right. \\nonumber \\\\
\n&{}& \\left.-\\bmr_l\\sin^2\\frac{\\theta}{2}\\right]\\bmi \\nonumber \\\\
\n&=& w+\\left[(\\bmu\\times\\bmr_l)\\sin\\theta+\\bmr_u+\\bmr_l\\cos\\theta\\right]\\bmi \\label{eq:result}
\n\\end{eqnarray}<\/p>\n

\u3068\u5909\u5f62\u3067\u304d\u307e\u3059\u3002<\/p>\n

\u3053\u3053\u3067$\\boldsymbol{r}_l$\u306f$\\boldsymbol{u}$\u3068\u76f4\u4ea4\u3059\u308b\u306e\u3067\u3001$||\\boldsymbol {u}\\times\\boldsymbol{r}_l|| = || \\boldsymbol{r}_l ||$\u3068\u306a\u308a\u3001\u304b\u3064$\\boldsymbol {u}\\times\\boldsymbol{r}_l$\u3068$\\boldsymbol{r}_l$\u306f\u76f4\u4ea4\u3059\u308b\u306e\u3067\u3001(\\ref{eq:result})\u5f0f\u53f3\u8fba\u7b2c2\u5f0f\u306e\u865a\u6570\u90e8\u306e\u4fc2\u6570(\u30d9\u30af\u30c8\u30eb)\u306f\u4ee5\u4e0b\u306e1.\u53ca\u30732.\u3092\u6e80\u305f\u3059\u3053\u3068\u304c\u308f\u304b\u308a\u307e\u3059\u3002<\/p>\n

    \n
  1. $\\boldsymbol{r}_u$\u3092\u4e2d\u5fc3\u3068\u3057\u3001$\\boldsymbol{u}\\times\\boldsymbol{r}_l$\u3068$\\boldsymbol{r}_l$\u304c\u5f35\u308b\u5e73\u9762\u4e0a\u306b\u3042\u308b\u5186\u4e0a\u306e\u70b9\u306b\u539f\u70b9\u304b\u3089\u5411\u304b\u3046\u30d9\u30af\u30c8\u30eb\u3067\u3042\u308b\u3053\u3068\u3002<\/li>\n
  2. 1.\u306e\u30d9\u30af\u30c8\u30eb\u306e\u7d42\u70b9\u306f$\\boldsymbol{r}$\u3092$\\boldsymbol{u}$\u306e\u307e\u308f\u308a\u306b$\\theta$\u3060\u3051\u56de\u8ee2\u3055\u305b\u305f\u70b9\u3067\u3042\u308b\u3053\u3068\u3002<\/li>\n<\/ol>\n

    \u3088\u3063\u3066\u3001(10)\u5f0f\u304c\u4e09\u6b21\u5143\u5ea7\u6a19\u7cfb\u306b\u304a\u3051\u308b\u56de\u8ee2\u5909\u63db\u3092\u8868\u73fe\u3057\u3066\u3044\u308b\u3053\u3068\u3092\u793a\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3057\u305f\u3002<\/p>\n

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

    \u5b9f\u306f\u3053\u306e\u8a18\u4e8b\u3092\u66f8\u3044\u3066\u3044\u308b\u9593\u306b\u56db\u5143\u6570\u306b\u3064\u3044\u3066\u306f(\u30d7\u30ed\u30b0\u30e9\u30df\u30f3\u30b0\u7b49\u3082\u542b\u3081\u3066)\u3059\u3067\u306b\u8a73\u3057\u3044\u8a18\u4e8b\u304c\u3044\u304f\u3064\u304b\u3042\u308b(\u300c\u53c2\u8003\u6587\u732e\u300d\u53c2\u7167\u3002)\u3053\u3068\u304c\u308f\u304b\u3063\u305f\u306e\u3067\u3001\u3067\u304d\u308b\u3060\u3051\u5225\u306e\u8996\u70b9\u3068\u3044\u3046\u304b\u3001\u8a08\u7b97\u306e\u8a73\u7d30\u306a\u904e\u7a0b\u3092\u66f8\u3044\u3066\u307f\u307e\u3057\u305f\u3002<\/p>\n

    \u3053\u306e\u8a18\u4e8b\u3067\u306f$w \\in \\mathbb{R}$\u3068\u304a\u3044\u3066\u304b\u3089\u8a08\u7b97\u3092\u884c\u3044\u307e\u3057\u305f\u3002\u5b9f\u969b\u306b(10)\u5f0f\u306e\u8a08\u7b97\u3092\u884c\u3046\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f5c\u6210\u3059\u308b\u5834\u5408\u306b\u306f\u3001$w=0$\u3068\u304a\u304f\u3068$w$\u304c\u73fe\u308c\u308b\u9805\u306e\u8a08\u7b97\u306f\u3057\u306a\u304f\u3066\u3082\u3088\u304f\u306a\u308a\u307e\u3059\u306e\u3067\u3001\u5c11\u3057\u3060\u3051\u8a08\u7b97\u91cf\u3092\u6e1b\u3089\u3059\u3053\u3068\u304c\u3067\u304d\u308b\u304b\u3082\u3057\u308c\u307e\u305b\u3093\u3002<\/p>\n

    \u3054\u89a7\u3044\u305f\u3060\u3044\u305f\u65b9\u306e\u304a\u5f79\u306b\u7acb\u3064\u304b\u3069\u3046\u304b\u306f\u308f\u304b\u308a\u307e\u305b\u3093\u304c\u3001\u53c2\u8003\u306b\u3057\u3066\u3044\u305f\u3060\u3051\u308b\u3068\u5e78\u3044\u3067\u3059\u3002<\/p>\n

    \u3053\u306e\u8a18\u4e8b\u306f\u4ee5\u4e0a\u3067\u3059\u304c\u3001Scala\u3067\u5b9f\u88c5\u3057\u3066\u307f\u307e\u3057\u305f\u306e\u3067\u3001\u3064\u3044\u3067\u306b\u8aad\u3093\u3067\u3044\u3063\u3066\u3044\u305f\u3060\u3051\u308b\u3068\u5e78\u3044\u3067\u3059(\u30ea\u30f3\u30af\u306f\u3053\u3061\u3089<\/a>)\u3002<\/p>\n

    \u53c2\u8003\u6587\u732e<\/h2>\n