\u5728\u8ba8\u8bba\u4e86\u4e8c\u7ef4\u65b9\u5dee\u5206\u5272\u4e4b\u540e\uff0c\u7ee7\u7eed\u8ba8\u8bba\u4e09\u7ef4\u65b9\u5dee\u5206\u5272\u7684\u95ee\u9898\uff1a\u5982\u4f55\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u9009\u62e9\u4e00\u6761\u76f4\u7ebf\\(l\\)\uff0c\u4f7f\u5f97\u4e09\u7ef4\u70b9\u96c6\\(A=\\{(x_i,y_i,z_i)|i=1,2,…N\\}\\)\u7684\u6295\u5f71\u70b9\\(P=\\{ p_i\u2208l|i=1,2,…N\\}\\)\u7684\u8ddd\u79bb\u65b9\u5dee\u6700\u5927\u3002<\/p>\n\n\n\n
\u901a\u8fc7\u524d\u9762\u5f97\u8ba8\u8bba\u53ef\u77e5\uff1a\u53ef\u4ee5\u4e09\u4f4d\u70b9\u96c6\\(A\\)\u7684\u8d28\u5fc3\u4e3a\u539f\u70b9\uff0c\u5efa\u7acb\u4e09\u7ef4\u76f4\u89d2\u5750\u6807\u7cfb\u3002\u8bbe\u76f4\u7ebf\\(l\\)\u901a\u8fc7\u5750\u6807\u539f\u70b9\uff0c\u5176\u5355\u4f4d\u65b9\u5411\u77e2\u91cf\\(\\overrightarrow l = (a,b,c)\\)\u3002\u4ee4\uff1a<\/p>\n\n\n\n
\\[H = \\sum\\limits_{i = 1}^N {{{(\\overrightarrow {O{p_i}} \\cdot \\overrightarrow l )}^2}} = \\sum\\limits_{i = 1}^N {{{(a{x_i} + b{y_i} + c{z_i})}^2}} \\]<\/p>\n\n\n\n
\u663e\u7136\\(H\\)\u662f\u5173\u4e8e\\(a,b,c\\)\u7684\u591a\u5143\u51fd\u6570\uff0c\u9644\u52a0\u6761\u4ef6\u4e3a\uff1a<\/p>\n\n\n\n
\\[ \\phi (a,b,c) = {a^2} + {b^2} + {c^2} – 1 = 0 \\]<\/p>\n\n\n\n
\u5148\u505a\u62c9\u683c\u6717\u65e5\u51fd\u6570\uff1a<\/p>\n\n\n\n
\\[ L(a,b,c) = H(a,b,c) + \\lambda \\cdot \\phi (a,b,c) \\]<\/p>\n\n\n\n
\u5176\u4e2d\\(\\lambda\\)\u4e3a\u53c2\u6570\uff0c\u5206\u522b\u5bf9\u53c2\u6570\\(a,b,c\\)\u6c42\u4e00\u9636\u504f\u5bfc\u6570\uff0c\u5e76\u4f7f\u4e4b\u7b49\u4e8e0\u3002\u7136\u540e\u8054\u7acb\u65b9\u7a0b\u7ec4\uff1a<\/p>\n\n\n\n
\\[ \\frac{{\\partial L}}{{\\partial a}} = \\frac{{\\partial H}}{{\\partial a}} + \\lambda \\cdot \\frac{{\\partial \\phi }}{{\\partial a}} = \\sum\\limits_{i = 1}^N {2(a{x_i} + b{y_i} + c{z_i}){x_i} + 2\\lambda a = 0} \\]<\/p>\n\n\n\n
\\[ \\frac{{\\partial L}}{{\\partial b}} = \\frac{{\\partial H}}{{\\partial b}} + \\lambda \\cdot \\frac{{\\partial \\phi }}{{\\partial b}} = \\sum\\limits_{i = 1}^N {2(a{x_i} + b{y_i} + c{z_i}){y_i} + 2\\lambda b = 0} \\]<\/p>\n\n\n\n
\\[\\frac{{\\partial L}}{{\\partial c}} = \\frac{{\\partial H}}{{\\partial c}} + \\lambda \\cdot \\frac{{\\partial \\phi }}{{\\partial c}} = \\sum\\limits_{i = 1}^N {2(a{x_i} + b{y_i} + c{z_i}){z_i} + 2\\lambda c = 0} \\]<\/p>\n\n\n\n
\u4ee4\uff1a<\/p>\n\n\n\n
\\[\\begin{array}{*{20}{c}}
{{m_{xx}} = \\sum\\limits_{i = 1}^N {{x_i}{x_i}} }&{{m_{xy}} = \\sum\\limits_{i = 1}^N {{x_i}{y_i}} }&{{m_{xz}} = \\sum\\limits_{i = 1}^N {{x_i}{z_i}} }
\\end{array}\\]<\/p>\n\n\n\n
\\[\\begin{array}{*{20}{c}}
{{m_{yy}} = \\sum\\limits_{i = 1}^N {{y_i}{y_i}} }&{{m_{yz}} = \\sum\\limits_{i = 1}^N {{y_i}{z_i}} }&{{m_{zz}} = \\sum\\limits_{i = 1}^N {{z_i}{z_i}} }
\\end{array}\\]<\/p>\n\n\n\n
\u6700\u540e\u53ef\u4ee5\u7b80\u5316\u5f97\u5230\u4e00\u4e2a\u4e00\u5143\u4e09\u6b21\u65b9\u7a0b\uff08\u8be5\u65b9\u7a0b\u8017\u8d39\u4e86\u4f5c\u8005\u4e0d\u5c11\u624b\u5de5\u7b97\u529b\uff09\uff1a<\/p>\n\n\n\n
\\[ {\\lambda ^3} + ({m_{xx}} + {m_{yy}} + {m_{zz}}){\\lambda ^2} \\\\ + ({m_{xx}}{m_{yy}} + {m_{xx}}{m_{zz}} + {m_{yy}}{m_{zz}} – m_{xy}^2 – m_{xz}^2 – m_{yz}^2)\\lambda \\\\ + ({m_{xx}}{m_{yy}}{m_{zz}} – {m_{xx}}m_{yz}^2 – {m_{yy}}m_{xz}^2 – {m_{zz}}m_{xy}^2 + 2{m_{xy}}{m_{xz}}{m_{yz}}) = 0 \\]<\/p>\n\n\n\n
\u5230\u6b64\u65f6\u53ef\u4ee5\u5229\u7528Matlab\u7b49\u8f6f\u4ef6\u76f4\u63a5\u6c42\u5f97\\(\\lambda\\)\u6570\u503c\u89e3\u3002\u5e26\u5165\u4e0a\u8ff0\u65b9\u7a0b\uff0c\u5373\u53ef\u89e3\u5f00\\(a,b,c\\)\u7684\u6570\u503c\u3002\u90a3\u4e48\u76f4\u7ebf\\(l\\)\u6240\u4ee3\u8868\u7684\u65b9\u5411\u662f\u8ddd\u79bb\u65b9\u5dee\u6700\u5927\u503c\u65b9\u5411\uff0c\u4e5f\u662f\u5206\u5272\u5e73\u9762\u7684\u6cd5\u77e2\u65b9\u5411\uff0c\u4e14\u8be5\u5206\u5272\u5e73\u9762\u8fc7\u8d28\u5fc3\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"
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