\u524d\u7de8\u304b\u3089\u5f15\u304d\u7d9a\u3044\u3066\u5206\u5b50\u306e\u56de\u8ee2\u306b\u3064\u3044\u3066\u3067\u3059\u3002\u4eca\u56de\u306f\u5206\u5b50\u3092\u91cd\u306d\u5408\u308f\u305b\u308b\u305f\u3081\u306e\u6570\u5f0f\u3092\u5c0e\u51fa\u3057\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002<\/p>\n\n\n\n
\u56db\u5143\u6570\u306e\u30a4\u30f3\u30c8\u30ed\u304c\u7d42\u308f\u3063\u3066\u3088\u3046\u3084\u304f\u672c\u984c\u306b\u5165\u308a\u307e\u3059\u3002\u5206\u5b50\u306e3\u6b21\u5143\u7684\u306a\u91cd\u306d\u5408\u308f\u305b\u3067\u3059\u304c\u3001\u52d5\u304b\u3059\u5206\u5b50\u306e\u5ea7\u6a19\u30d9\u30af\u30c8\u30eb\u3092\\( \\mathbf{r}_{\\alpha}=(x_{\\alpha}, y_{\\alpha}, z_{\\alpha}) \\)\u3068\u3057\u3066\u3001\u91cd\u306d\u308b\u5148\u306e\u53c2\u7167\u5206\u5b50\u306e\u5ea7\u6a19\u30d9\u30af\u30c8\u30eb\u3092\\( \\mathbf{r}_{0\\alpha}=(x_{0\\alpha}, y_{0\\alpha}, z_{0\\alpha}) \\)\u3068\u3057\u307e\u3059\u3002\u3053\u3053\u3067\\( \\alpha \\)\u306f\u539f\u5b50\u306e\u30a4\u30f3\u30c7\u30c3\u30af\u30b9\u3068\u3057\u307e\u3059\u3002\u5206\u5b50\u3092\u52d5\u304b\u3057\u3066\u53c2\u7167\u5206\u5b50\u306b\u91cd\u306d\u308b\u3068\u3044\u3046\u3053\u3068\u306f\u30012\u3064\u306e\u5206\u5b50\u306e\u5ea7\u6a19\u306e\u5dee\u3092\u6700\u5c0f\u306b\u3059\u308b\u3068\u3044\u3046\u554f\u984c\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
\u307e\u305a\u5206\u5b50\u5ea7\u6a19\u306e\u6b8b\u5dee\u30d9\u30af\u30c8\u30eb\u306f\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u306a\u308a\u307e\u3059\u3002
$$
\\mathbf{\\epsilon}_{\\alpha} = R\\left(\\mathbf{r}_{\\alpha} – \\mathbf{g}\\right) – \\left(\\mathbf{r}_{0\\alpha}-\\mathbf{g}_{0}\\right)
$$
\u3053\u3053\u3067\\(R\\)\u306f\u56de\u8ee2\u884c\u5217\u3067\u3001\\(\\mathbf{g}\\)\u3068\\(\\mathbf{g}_{0}\\)\u306f\u305d\u308c\u305e\u308c\u52d5\u304b\u3059\u5206\u5b50\u3068\u53c2\u7167\u5206\u5b50\u306e\u6bcd\u6838\u91cd\u5fc3\uff08\u4e2d\u5fc3\uff09\u30d9\u30af\u30c8\u30eb\u3068\u306a\u308a\u307e\u3059\u3002 2\u3064\u306e\u5206\u5b50\u306e\u6bcd\u6838\u91cd\u5fc3\u3092\u539f\u70b9\u306b\u79fb\u52d5\u3055\u305b\u308b\u3053\u3068\u306f\u7c21\u5358\u306a\u3053\u3068\u3067\u3059\u3002\u4e88\u3081\u52d5\u304b\u3059\u5206\u5b50\u3068\u53c2\u7167\u5206\u5b50\u306e\u6bcd\u6838\u91cd\u5fc3\u306f\u5ea7\u6a19\u7cfb\u306e\u539f\u70b9\u306b\u4e00\u81f4\u3057\u3066\u3044\u308b\u3082\u306e\u3068\u3059\u308c\u3070\u3001 \u4ee5\u5f8c\u306f\u7d14\u7c8b\u306b\u56de\u8ee2\u306e\u307f\u3092\u8003\u3048\u308c\u3070\u3088\u3044\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002 \u3057\u305f\u304c\u3063\u3066\u4ee5\u5f8c\u306e\u8b70\u8ad6\u3067\u306f\u52d5\u304b\u3059\u5206\u5b50\u3068\u53c2\u7167\u5206\u5b50\u306e\u6bcd\u6838\u91cd\u5fc3\u306f\u539f\u70b9\u306b\u3042\u308b\u3082\u306e\u3068\u3057\u3066
$$
\\mathbf{g}=(0,0,0) \\\\
\\mathbf{g}_{0}=(0,0,0)
$$
\u3067\u3042\u308b\u3053\u3068\u3092\u4eee\u5b9a\u3057\u307e\u3059\u3002\u305d\u306e\u3046\u3048\u3067\u4e0a\u8a18\u306e\u6b8b\u5dee\u3092\u56db\u5143\u6570\u3092\u4f7f\u3063\u3066\u8868\u73fe\u3059\u308b\u3068\u3001
$$
\\epsilon_{\\alpha} = q r_{\\alpha} \\bar{q} – r_{0\\alpha}
$$
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u6b8b\u5dee\u3092\u6bcd\u6838\u3092\u69cb\u6210\u3059\u308b\u539f\u5b50\u306b\u5168\u3066\u306b\u304a\u3044\u3066\u6700\u5c0f\u5316\u3057\u305f\u3044\u306e\u3067\u3001\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u95a2\u6570\u3092\u6700\u5c0f\u5316\u3059\u308b\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002
$$
m(q) = \\displaystyle\\sum_{\\alpha} \\bar{\\epsilon_{\\alpha}} \\epsilon_{\\alpha}
$$
\u3053\u306e\u95a2\u6570\u3092\u6700\u5c0f\u5316\u3059\u308b\u3088\u3046\u306a\u56db\u5143\u6570\\(q\\)\u3092\u6c42\u3081\u308c\u3070\u826f\u3044\u306e\u3067\u3059\u3002\u305f\u3060\u3057\u3001\u56de\u8ee2\u3092\u8868\u3059\u56db\u5143\u6570\u306f\u5927\u304d\u3055\u304c1\u306a\u306e\u3067\u3001\\(q^2=\\bar{q}q=1\\)\u3068\u3044\u3046\u6761\u4ef6\u3082\u8ab2\u3059\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
\u4e0a\u8a18\u306e\u6b8b\u5dee\u56db\u5143\u6570\u306e\u307e\u307e\u3060\u3068\u8a08\u7b97\u304c\u9762\u5012\u306a\u306e\u3067\u3001\u5c11\u3057\u5909\u5f62\u3057\u307e\u3059\u3002\\(\\epsilon_{\\alpha}\\)\u306e\u53f3\u304b\u3089\\(q\\)\u3092\u639b\u3051\u305f\u3082\u306e\u3092\u8003\u3048\u307e\u3059\u3002
$$
\\epsilon_{\\alpha}^{\\prime}=\\epsilon_{\\alpha}q = q r_{\\alpha} – r_{0\\alpha}q
$$
\u305d\u3057\u3066\u76ee\u7684\u95a2\u6570\u306f\u6b21\u306e\u3088\u3046\u306a\u5f62\u306b\u306a\u308a\u307e\u3059\u3002
$$
\\begin{align}
m^{\\prime}(q) &= \\displaystyle\\sum_{\\alpha} \\bar{\\epsilon_{\\alpha}^{\\prime}} \\epsilon_{\\alpha}^{\\prime} \\\\
&= \\overline{(q r_{\\alpha} – r_{0\\alpha}q)}(q r_{\\alpha} – r_{0\\alpha}q)
\\end{align}
$$
\u3053\u3053\u3067\\(q^2=\\bar{q}q=1\\)\u3068\u3044\u3046\u6761\u4ef6\u3092\u4f7f\u3046\u3068\\(m^{\\prime}(q) = m(q)\\)\u3068\u306a\u308a\u307e\u3059\u3002\u305c\u3072\u624b\u3092\u52d5\u304b\u3057\u3066\u78ba\u8a8d\u3057\u3066\u307f\u3066\u304f\u3060\u3055\u3044\u3002<\/p>\n\n\n\n
\u56db\u5143\u6570\\(q=a+bi+cj+dk\\)\u30924\u6b21\u5143\u30d9\u30af\u30c8\u30eb\\(\\mathbf{\\tilde{q}}=(a,b,c,d)\\)\u3068\u3057\u3066\u3001\u3068\u3042\u308b\\(4\\times4\\)\u306e\u884c\u5217\\(K_{\\alpha}\\)\u3092\u4f7f\u3063\u3066\u4e0a\u8a18\u76ee\u7684\u95a2\u6570\u306f\u3001
$$
m^{\\prime}(q) = \\displaystyle\\sum_{\\alpha}
\\mathbf{\\tilde{q}}^{\\mathrm{T}}K_{\\alpha}^{\\mathrm{T}}K_{\\alpha}\\mathbf{\\tilde{q}}
= \\displaystyle\\sum_{\\alpha} \\mathbf{\\tilde{q}}^{\\mathrm{T}}M_{\\alpha}\\mathbf{\\tilde{q}}
$$
\u3068\u66f8\u304d\u8868\u3059\u3053\u3068\u304c\u3067\u304d\u307e\u3059\u3002\u884c\u5217\\(M_{\\alpha}=K_{\\alpha}^{\\mathrm{T}}K_{\\alpha}\\)\u306f\u5bfe\u79f0\u884c\u5217\u3068\u306a\u308a\u307e\u3059\u3002\u305d\u3057\u3066\u56db\u5143\u6570\\(q\\)\u306e\u5927\u304d\u3055\u304c1\u3067\u3042\u308b\u3068\u3044\u3046\u6761\u4ef6\u306f\u3001
$$
\\mathbf{\\tilde{q}}^{\\mathrm{T}}\\mathbf{\\tilde{q}} = 1
$$
\u3068\u306a\u308a\u307e\u3059\u3002\u3057\u305f\u304c\u3063\u3066\u3001\u3053\u306e\u6761\u4ef6\u306e\u4e0b\u306b\\(m^{\\prime}(q)\\)\u3092\u6700\u5c0f\u5316\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u306f\u3001
$$
L=\\displaystyle\\sum_{\\alpha} \\mathbf{\\tilde{q}}^{\\mathrm{T}}M_{\\alpha}\\mathbf{\\tilde{q}}
-\\lambda \\left( \\mathbf{\\tilde{q}}^{\\mathrm{T}}\\mathbf{\\tilde{q}} – 1 \\right)
$$
\u3092\u6700\u5c0f\u5316\u3059\u308b\u3068\u3044\u3046\u3053\u3068\u306b\u306a\u308a\u307e\u3059\u3002\u3053\u3053\u3067\\(\\lambda\\)\u306f\u30e9\u30b0\u30e9\u30f3\u30b8\u30e5\u306e\u672a\u5b9a\u4e57\u6570\u3067\u3059\u3002\u6700\u7d42\u7684\u306b\u4ee5\u4e0b\u306e\u3088\u3046\u306a\u56fa\u6709\u5024\u554f\u984c\u306b\u5e30\u7740\u3057\u307e\u3059\u3002
$$
\\displaystyle\\sum_{\\alpha} M_{\\alpha}\\mathbf{\\tilde{q}}
= M\\mathbf{\\tilde{q}} = \\lambda \\mathbf{\\tilde{q}} \\\\
M=\\displaystyle\\sum_{\\alpha} M_{\\alpha}
$$
4\u6b21\u306e\u6b63\u65b9\u5bfe\u79f0\u884c\u5217\\(M\\)\u3092\u5bfe\u89d2\u5316\u3057\u3066\u56fa\u6709\u5024\u6700\u5c0f\u306e\u56fa\u6709\u30d9\u30af\u30c8\u30eb\\(\\mathbf{\\tilde{q}}=(a, b, c, d)\\)\u3092\u6c42\u3081\u308c\u3070\u3001\u56db\u5143\u6570\\(q=a+bi+cj+dk\\)\u304b\u3089\u56de\u8ee2\u884c\u5217\\(R\\)\u304c\u8a08\u7b97\u3067\u304d\u308b\u306e\u3067\u3001\u53c2\u7167\u5206\u5b50\u306e\u6bcd\u6838\u69cb\u9020\u306b\u30d4\u30c3\u30bf\u30ea\u91cd\u306a\u308b\u3088\u3046\u306b\u5206\u5b50\u3092\u56de\u8ee2\u3067\u304d\u308b\u306f\u305a\u3067\u3059\u3002\u6b21\u306f\\(M_{\\alpha}\\)\u306e\u5177\u4f53\u7684\u306a\u8868\u5f0f\u3092\u8a08\u7b97\u3057\u3066\u307f\u307e\u3057\u3087\u3046\u3002<\/p>\n\n\n\n
\\(M_{\\alpha}\\)\u3092\u8a08\u7b97\u3059\u308b\u305f\u3081\u306b\u3001\u5148\u305a\u306f\u4ee5\u4e0b\u306e\u5f0f\u3092\u8a08\u7b97\u3057\u3066\\(K_{\\alpha}\\)\u307f\u307e\u3057\u3087\u3046\u3002
$$
\\epsilon_{\\alpha}^{\\prime} = q r_{\\alpha} – r_{0\\alpha}q
= K_{\\alpha} \\mathbf{\\tilde{q}}
$$
\u56db\u5143\u6570\\(q, r_{\\alpha}, r_{0\\alpha}\\)\u306e\u30d9\u30af\u30c8\u30eb\u8868\u73fe\u306f\u305d\u308c\u305e\u308c\u4ee5\u4e0b\u306e\u901a\u308a\u3067\u3059\u3002
$$
\\begin{align}
&q = (a, \\mathbf{q}) = (a, (b, c, d)) \\\\
&r_{\\alpha} = (0, \\mathbf{r}_{\\alpha}) = (0, (x_{\\alpha}, y_{\\alpha}, z_{\\alpha})) \\\\
&r_{0\\alpha} = (0, \\mathbf{r}_{0\\alpha}) = (0, (x_{0\\alpha}, y_{0\\alpha}, z_{0\\alpha}))
\\end{align}
$$
\u524d\u7de8\u3067\u7d39\u4ecb\u3057\u305f\u30d9\u30af\u30c8\u30eb\u8868\u73fe\u3067\u306e\u7a4d\u306e\u516c\u5f0f\u3092\u4f7f\u3046\u3068\u3001
$$
\\begin{align}
&q r_{\\alpha} = (-\\mathbf{q} \\cdot \\mathbf{r}_{\\alpha}, a\\mathbf{r}_{\\alpha} + \\mathbf{q} \\times \\mathbf{r}_{\\alpha}) \\\\ &r_{0\\alpha} q = (-\\mathbf{r}_{0\\alpha} \\cdot \\mathbf{q}, a\\mathbf{r}_{0\\alpha} + \\mathbf{r}_{0\\alpha} \\times \\mathbf{q})
\\end{align}
$$
2\u3064\u306e\u5dee\u3092\u3068\u3063\u3066\u3001
$$
\\begin{align}
q r_{\\alpha} – r_{0\\alpha} q
&= ((\\mathbf{r}_{0\\alpha}-\\mathbf{r}_{\\alpha}) \\cdot \\mathbf{q}, \\quad
-a(\\mathbf{r}_{0\\alpha}-\\mathbf{r}_{\\alpha}) + \\mathbf{q} \\times (\\mathbf{r}_{0\\alpha}+\\mathbf{r}_{\\alpha}) )\\\\
&=\\begin{pmatrix}
(x_{0\\alpha}-x_{\\alpha})b+(y_{0\\alpha}-y_{\\alpha})c+(z_{0\\alpha}-z_{\\alpha})d \\\\
-(x_{0\\alpha}-x_{\\alpha})a+(z_{0\\alpha}+z_{\\alpha})c-(y_{0\\alpha}+y_{\\alpha})d \\\\
-(y_{0\\alpha}-y_{\\alpha})a-(z_{0\\alpha}+z_{\\alpha})b+(x_{0\\alpha}+x_{\\alpha})d \\\\
-(z_{0\\alpha}-z_{\\alpha})a+(y_{0\\alpha}+y_{\\alpha})b-(x_{0\\alpha}+x_{\\alpha})c
\\end{pmatrix} \\\\
&=\\begin{pmatrix}
0 & x_{0\\alpha}-x_{\\alpha} & y_{0\\alpha}-y_{\\alpha} & z_{0\\alpha}-z_{\\alpha} \\\\
-(x_{0\\alpha}-x_{\\alpha}) & 0 & z_{0\\alpha}+z_{\\alpha} & -(y_{0\\alpha}+y_{\\alpha}) \\\\
-(y_{0\\alpha}-y_{\\alpha}) & -(z_{0\\alpha}+z_{\\alpha}) & 0 & x_{0\\alpha}+x_{\\alpha} \\\\
-(z_{0\\alpha}-z_{\\alpha}) & y_{0\\alpha}+y_{\\alpha} & -(x_{0\\alpha}+x_{\\alpha}) & 0 \\\\
\\end{pmatrix}
\\begin{pmatrix} a \\\\ b \\\\ c \\\\ d \\end{pmatrix} \\\\
&= K_{\\alpha} \\mathbf{\\tilde{q}}
\\end{align}
$$
\u3053\u308c\u3067\u884c\u5217\\(K_{\\alpha}\\)\u304c\u6c42\u307e\u308a\u307e\u3057\u305f\u3002<\/p>\n\n\n\n
\u3044\u3088\u3044\u3088\u884c\u5217\\(M_{\\alpha}\\)\u3092\u8a08\u7b97\u3057\u307e\u3059\u3002
\u3053\u306e\u8a08\u7b97\u306f\u898b\u304b\u3051\u307b\u3069\u9762\u5012\u3067\u306f\u306a\u3044\u306e\u3067\u6839\u6027\u3067\u3084\u308a\u307e\u3057\u3087\u3046\u3002
$$
\\begin{align}
M_{\\alpha} &= K_{\\alpha}^{\\mathrm{T}} K_{\\alpha} \\\\
&=\\begin{pmatrix}
0 & -(x_{0\\alpha}-x_{\\alpha}) & -(y_{0\\alpha}-y_{\\alpha}) & -(z_{0\\alpha}-z_{\\alpha}) \\\\
x_{0\\alpha}-x_{\\alpha} & 0 & -(z_{0\\alpha}+z_{\\alpha}) & y_{0\\alpha}+y_{\\alpha} \\\\
y_{0\\alpha}-y_{\\alpha} & z_{0\\alpha}+z_{\\alpha} & 0 & -(x_{0\\alpha}+x_{\\alpha}) \\\\
z_{0\\alpha}-z_{\\alpha} & -(y_{0\\alpha}+y_{\\alpha}) & x_{0\\alpha}+x_{\\alpha} & 0 \\\\
\\end{pmatrix}
\\begin{pmatrix}
0 & x_{0\\alpha}-x_{\\alpha} & y_{0\\alpha}-y_{\\alpha} & z_{0\\alpha}-z_{\\alpha} \\\\
-(x_{0\\alpha}-x_{\\alpha}) & 0 & z_{0\\alpha}+z_{\\alpha} & -(y_{0\\alpha}+y_{\\alpha}) \\\\
-(y_{0\\alpha}-y_{\\alpha}) & -(z_{0\\alpha}+z_{\\alpha}) & 0 & x_{0\\alpha}+x_{\\alpha} \\\\
-(z_{0\\alpha}-z_{\\alpha}) & y_{0\\alpha}+y_{\\alpha} & -(x_{0\\alpha}+x_{\\alpha}) & 0 \\\\
\\end{pmatrix} \\\\
&=\\begin{pmatrix}
m_{\\alpha 11} & m_{\\alpha 12} & m_{\\alpha 13} & m_{\\alpha 14} \\\\
m_{\\alpha 21} & m_{\\alpha 22} & m_{\\alpha 23} & m_{\\alpha 14} \\\\
m_{\\alpha 31} & m_{\\alpha 32} & m_{\\alpha 33} & m_{\\alpha 14} \\\\
m_{\\alpha 41} & m_{\\alpha 42} & m_{\\alpha 43} & m_{\\alpha 44} \\\\
\\end{pmatrix}
\\end{align}
$$
\u5f0f\u304c\u9577\u304f\u3066\u884c\u5217\u306e\u5f62\u3067\u66f8\u3051\u306a\u3044\u306e\u3067\u3001\u884c\u5217\u6210\u5206\u6bce\u306b\u66f8\u304d\u4e0b\u3057\u307e\u3059\u3002
\u307e\u305f\u3001\\(M_{\\alpha}\\)\u306f\u5bfe\u79f0\u884c\u5217\u306a\u306e\u3067\u4e0b\u534a\u5206\u3060\u3051\u306e\u8a18\u8ff0\u3068\u3057\u307e\u3059\u3002
$$
\\begin{align}
m_{\\alpha 11} &= x_{\\alpha}^2+y_{\\alpha}^2+z_{\\alpha}^2+x_{0\\alpha}^2+y_{0\\alpha}^2+z_{0\\alpha}^2
-2x_{\\alpha}x_{0\\alpha}-2y_{\\alpha}y_{0\\alpha}-2z_{\\alpha}z_{0\\alpha} \\\\
m_{\\alpha 21} &= 2(z_{\\alpha}y_{0\\alpha}-y_{\\alpha}z_{0\\alpha}) \\\\
m_{\\alpha 22} &= x_{\\alpha}^2+y_{\\alpha}^2+z_{\\alpha}^2+x_{0\\alpha}^2+y_{0\\alpha}^2+z_{0\\alpha}^2
-2x_{\\alpha}x_{0\\alpha}+2y_{\\alpha}y_{0\\alpha}+2z_{\\alpha}z_{0\\alpha} \\\\
m_{\\alpha 31} &= 2(x_{\\alpha}z_{0\\alpha}-z_{\\alpha}x_{0\\alpha}) \\\\
m_{\\alpha 32} &= -2(x_{\\alpha}y_{0\\alpha}+y_{\\alpha}x_{0\\alpha}) \\\\
m_{\\alpha 33} &= x_{\\alpha}^2+y_{\\alpha}^2+z_{\\alpha}^2+x_{0\\alpha}^2+y_{0\\alpha}^2+z_{0\\alpha}^2
+2x_{\\alpha}x_{0\\alpha}-2y_{\\alpha}y_{0\\alpha}+2z_{\\alpha}z_{0\\alpha} \\\\
m_{\\alpha 41} &= 2(y_{\\alpha}x_{0\\alpha}-x_{\\alpha}y_{0\\alpha}) \\\\
m_{\\alpha 42} &= -2(x_{\\alpha}z_{0\\alpha}+z_{\\alpha}x_{0\\alpha}) \\\\
m_{\\alpha 43} &= -2(y_{\\alpha}z_{0\\alpha}+z_{\\alpha}y_{0\\alpha}) \\\\
m_{\\alpha 44} &= x_{\\alpha}^2+y_{\\alpha}^2+z_{\\alpha}^2+x_{0\\alpha}^2+y_{0\\alpha}^2+z_{0\\alpha}^2
+2x_{\\alpha}x_{0\\alpha}+2y_{\\alpha}y_{0\\alpha}-2z_{\\alpha}z_{0\\alpha} \\\\
\\end{align}
$$
\u884c\u5217\\(M\\)\u306f\u91cd\u306d\u5408\u308f\u305b\u306b\u4f7f\u3046\u539f\u5b50\u306b\u6e21\u3063\u3066\\(M_{\\alpha}\\)\u306e\u548c\u3092\u3068\u308c\u3070\u826f\u3044\u306e\u3067\u3001
$$
\\scriptsize
\\begin{align}
M &= \\displaystyle\\sum_{\\alpha} M_{\\alpha} \\\\
&=\\begin{pmatrix}
\\displaystyle\\sum_{\\alpha} \\left( d_{\\alpha}^2-2(x_{\\alpha}x_{0\\alpha}+y_{\\alpha}y_{0\\alpha}+z_{\\alpha}z_{0\\alpha}) \\right)&
\\displaystyle\\sum_{\\alpha} 2(z_{\\alpha}y_{0\\alpha}-y_{\\alpha}z_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} 2(x_{\\alpha}z_{0\\alpha}-z_{\\alpha}x_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} 2(y_{\\alpha}x_{0\\alpha}-x_{\\alpha}y_{0\\alpha}) \\\\
\\displaystyle\\sum_{\\alpha} 2(z_{\\alpha}y_{0\\alpha}-y_{\\alpha}z_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} \\left( d_{\\alpha}^2+2(-x_{\\alpha}x_{0\\alpha}+y_{\\alpha}y_{0\\alpha}+z_{\\alpha}z_{0\\alpha}) \\right)&
\\displaystyle\\sum_{\\alpha} -2(x_{\\alpha}y_{0\\alpha}+y_{\\alpha}x_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} -2(x_{\\alpha}z_{0\\alpha}+z_{\\alpha}x_{0\\alpha}) \\\\
\\displaystyle\\sum_{\\alpha} 2(x_{\\alpha}z_{0\\alpha}-z_{\\alpha}x_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} -2(x_{\\alpha}y_{0\\alpha}+y_{\\alpha}x_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} \\left( d_{\\alpha}^2+2(x_{\\alpha}x_{0\\alpha}-y_{\\alpha}y_{0\\alpha}+z_{\\alpha}z_{0\\alpha}) \\right)&
\\displaystyle\\sum_{\\alpha} -2(y_{\\alpha}z_{0\\alpha}+z_{\\alpha}y_{0\\alpha}) \\\\
\\displaystyle\\sum_{\\alpha} 2(y_{\\alpha}x_{0\\alpha}-x_{\\alpha}y_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} -2(x_{\\alpha}z_{0\\alpha}+z_{\\alpha}x_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} -2(y_{\\alpha}z_{0\\alpha}+z_{\\alpha}y_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} \\left( d_{\\alpha}^2+2(x_{\\alpha}x_{0\\alpha}+y_{\\alpha}y_{0\\alpha}-z_{\\alpha}z_{0\\alpha}) \\right) \\\\
\\end{pmatrix}
\\\\
d_{\\alpha}^2&=x_{\\alpha}^2+y_{\\alpha}^2+z_{\\alpha}^2+x_{0\\alpha}^2+y_{0\\alpha}^2+z_{0\\alpha}^2
\\end{align}
$$
\u3068\u306a\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
\u5206\u5b50\u306e\u6bcd\u6838\u540c\u58eb\u3092\u91cd\u306d\u5408\u308f\u305b\u308b\u56de\u8ee2\u884c\u5217\\(R\\)\u3092\u6c42\u3081\u308b\u306b\u306f\u3001\u4e0a\u8a18\u306e\\(4 \\times 4\\)\u884c\u5217\\(M\\)\u3092\u5bfe\u89d2\u5316\u3057\u3066\u3001\u6700\u5c0f\u56fa\u6709\u5024\u306e\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u304b\u3089\u56db\u5143\u6570\u3092\u69cb\u6210\u3059\u308c\u3070\u826f\u3044\u3053\u3068\u306f\u5206\u304b\u308a\u307e\u3057\u305f\u3002\u884c\u5217\\(M\\)\u3092\u305d\u306e\u307e\u307e\u30d7\u30ed\u30b0\u30e9\u30df\u30f3\u30b0\u3057\u3066\u8a08\u7b97\u3057\u3066\u3082\u826f\u3044\u306e\u3067\u3059\u304c\u3001\u3082\u3046\u5c11\u3057\u30b7\u30f3\u30d7\u30eb\u306a\u5f62\u306b\u5909\u5f62\u3057\u305f\u3044\u3068\u601d\u3044\u307e\u3059\u3002\u884c\u5217\u3092\u3088\u308a\u30b7\u30f3\u30d7\u30eb\u306a\u5f62\u306b\u5909\u5f62\u3067\u304d\u308c\u3070\u30d7\u30ed\u30b0\u30e9\u30df\u30f3\u30b0\u3059\u308b\u3068\u304d\u306e\u30df\u30b9\u304c\u6e1b\u308b\u3060\u308d\u3046\u3057\u3001\u6570\u5024\u8aa4\u5dee\u3082\u8efd\u6e1b\u3067\u304d\u308b\u5834\u5408\u3082\u3042\u308a\u307e\u3059\u3002<\/p>\n\n\n\n
\u884c\u5217\\(M\\)\u3092\u4ee5\u4e0b\u306e\u3088\u3046\u306b\u5909\u5f62\u3057\u307e\u3059\u3002
$$
M = 2M^{\\prime}+\\displaystyle\\sum_{\\alpha}(x_{\\alpha}^2+y_{\\alpha}^2+z_{\\alpha}^2+x_{0\\alpha}^2+y_{0\\alpha}^2+z_{0\\alpha}^2)I
$$
\u3053\u3053\u3067\\(I\\)\u306f\\(4 \\times 4\\)\u306e\u5358\u4f4d\u884c\u5217\u3067\u3059\u3002\u305d\u3057\u3066\\(M^{\\prime}\\)\u306f
$$
M^{\\prime} = \\begin{pmatrix}
\\displaystyle\\sum_{\\alpha} (-x_{\\alpha}x_{0\\alpha}-y_{\\alpha}y_{0\\alpha}-z_{\\alpha}z_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} (z_{\\alpha}y_{0\\alpha}-y_{\\alpha}z_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} (x_{\\alpha}z_{0\\alpha}-z_{\\alpha}x_{0\\alpha})&
\\displaystyle\\sum_{\\alpha} (y_{\\alpha}x_{0\\alpha}-x_{\\alpha}y_{0\\alpha}) \\\\
\\displaystyle\\sum_{\\alpha} (z_{\\alpha}y_{0\\alpha}-y_{\\alpha}z_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} (-x_{\\alpha}x_{0\\alpha}+y_{\\alpha}y_{0\\alpha}+z_{\\alpha}z_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} (-x_{\\alpha}y_{0\\alpha}-y_{\\alpha}x_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} (-x_{\\alpha}z_{0\\alpha}-z_{\\alpha}x_{0\\alpha}) \\\\
\\displaystyle\\sum_{\\alpha} (x_{\\alpha}z_{0\\alpha}-z_{\\alpha}x_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} (-x_{\\alpha}y_{0\\alpha}-y_{\\alpha}x_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} (x_{\\alpha}x_{0\\alpha}-y_{\\alpha}y_{0\\alpha}+z_{\\alpha}z_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} (-y_{\\alpha}z_{0\\alpha}-z_{\\alpha}y_{0\\alpha}) \\\\
\\displaystyle\\sum_{\\alpha} (y_{\\alpha}x_{0\\alpha}-x_{\\alpha}y_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} (-x_{\\alpha}z_{0\\alpha}-z_{\\alpha}x_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} (-y_{\\alpha}z_{0\\alpha}-z_{\\alpha}y_{0\\alpha}) &
\\displaystyle\\sum_{\\alpha} (x_{\\alpha}x_{0\\alpha}+y_{\\alpha}y_{0\\alpha}-z_{\\alpha}z_{0\\alpha}) \\\\
\\end{pmatrix}
$$
\u3068\u306a\u308a\u307e\u3059\u3002\u3053\u306e\u884c\u5217\\(M^{\\prime}\\)\u306e\u56fa\u6709\u30d9\u30af\u30c8\u30eb\\(\\mathbf{u}\\)\u306f\u884c\u5217\\(M\\)\u306e\u56fa\u6709\u30d9\u30af\u30c8\u30eb\u3067\u3082\u3042\u308a\u307e\u3059\u3002
$$
\\begin{align}
M \\mathbf{u} &= 2M^{\\prime} \\mathbf{u} + \\displaystyle\\sum_{\\alpha}(x_{\\alpha}^2+y_{\\alpha}^2+z_{\\alpha}^2+x_{0\\alpha}^2+y_{0\\alpha}^2+z_{0\\alpha}^2)I \\mathbf{u} \\\\
&= \\left( 2\\lambda^{\\prime} + \\displaystyle\\sum_{\\alpha}(x_{\\alpha}^2+y_{\\alpha}^2+z_{\\alpha}^2+x_{0\\alpha}^2+y_{0\\alpha}^2+z_{0\\alpha}^2) \\right) \\mathbf{u} \\\\
&= \\lambda \\mathbf{u}
\\end{align}
$$
\u3057\u305f\u304c\u3063\u3066\u30d7\u30ed\u30b0\u30e9\u30df\u30f3\u30b0\u3059\u308b\u3068\u304d\u306f\u884c\u5217\\(M\\)\u3088\u308a\u3082\u30b7\u30f3\u30d7\u30eb\u306a\\(M^{\\prime}\\)\u3092\u7528\u3044\u308b\u3053\u3068\u306b\u3057\u307e\u3059\u3002<\/p>\n\n\n\n
\u6b21\u56de\u306f\u5b9f\u969b\u306bC++\u30d7\u30ed\u30b0\u30e9\u30e0\u3092\u4f5c\u3063\u3066\u307f\u307e\u3059\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
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