{"id":8007,"date":"2020-08-31T07:20:19","date_gmt":"2020-08-30T23:20:19","guid":{"rendered":"http:\/\/www.andrewsun.net\/panta_rhei\/?p=8007"},"modified":"2022-05-17T04:24:42","modified_gmt":"2022-05-16T20:24:42","slug":"%e6%ac%a7%e5%87%a0%e9%87%8c%e5%be%b7%e7%a9%ba%e9%97%b4%e7%9a%84%e7%8e%b0%e4%bb%a3%e5%bc%95%e5%85%a5%e4%b8%ad%e7%9a%84%e4%b8%80%e4%b8%aa%e9%97%ae%e9%a2%98","status":"publish","type":"post","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/8007","title":{"rendered":"\u6b27\u51e0\u91cc\u5fb7\u7a7a\u95f4\u7684\u73b0\u4ee3\u5f15\u5165\u4e2d\u7684\u4e00\u4e2a\u95ee\u9898"},"content":{"rendered":"<div class=\"kcite-section\" kcite-section-id=\"8007\">\n\n<p>[latexpage]<\/p>\n\n\n\n<p>$\\mathcal{E}$\u662f\u4e00\u4e2a\u975e\u7a7a\u96c6\u5408\uff0c\u91cc\u9762\u7684\u5143\u7d20\u79f0\u4e3a\u201c\u70b9\u201d\u3002\u5bf9\u4e8e\u96c6\u5408\u4e2d\u7684\u8fd9\u4e9b\u70b9\uff0c\u6211\u4eec\u53ef\u4ee5\u505a\u300a\u51e0\u4f55\u539f\u672c\u300b\u8bf4\u80fd\u505a\u7684\u6240\u6709\u4e8b\uff0c\u4ece\u800c\u5f97\u5230\u4e00\u5207\u300a\u51e0\u4f55\u539f\u672c\u300b\u80fd\u5f97\u5230\u7684\u7ed3\u8bba\u3002\u5176\u4e2d\u5c31\u5305\u62ec\u5141\u8bb8\u6211\u4eec\u8c08\u53ca\u201c\u7531\u70b9$a$\u5230\u70b9$b$\u7684\u6709\u5411\u7ebf\u6bb5\u201d\uff0c$a,b\\in\\mathcal{E}$\u3002\u6211\u4eec\u5efa\u7acb\u4e00\u4e2a\u201c\u4ece\u70b9$a$\u5230\u70b9$b$\u7684\u6709\u5411\u7ebf\u6bb5\u201d\u5230\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4$\\mathcal{V}$\u7684\u6620\u5c04$\\Phi:\\mathcal{E}\\times\\mathcal{E}\\rightarrow\\mathcal{V}$\uff0c\u4e14$\\Phi\\left(x,x\\right)\\equiv\\mathbf{0}\\forall x\\in\\mathcal{E}$\uff0c\u5176\u4e2d$\\mathbf{0}$\u662f$\\mathcal{V}$\u7684\u96f6\u5411\u91cf\u3002\u9009\u5b9a\u4e00\u4e2a\u70b9$o\\in\\mathcal{E}$\uff0c\u53c8\u53ef\u8bb0$\\Phi_o:\\mathcal{E}\\rightarrow\\mathcal{V}$\uff0c$\\Phi_o\\left(x\\right)\\equiv\\Phi\\left(o,x\\right)$\u3002\u5728\u5f88\u591a\u4e66\u4e2d<span id=\"90b329ff-cdfd-435c-b5ab-abccb630ad9f\" data-items=\"[&quot;1414259973&quot;,&quot;1278065742&quot;]\" contenteditable=\"false\" class=\"abt-citation\">\u200b(Berger 1987; Audin 2002)\u200b<\/span>\uff0c\u76f4\u63a5\u9884\u5148\u628a$\\Phi_o$\u89c4\u5b9a\u4e3a\u53cc\u5c04\u3002\u4f46\u6211\u5728\u8fd9\u91cc\u60f3\u5148\u4e0d\u4f5c\u6b64\u89c4\u5b9a\u3002\u6700\u591a\uff0c\u7531\u4e8e\u6211\u4eec\u5e0c\u671b$\\mathcal{E}$\u4e2d\u7684\u6bcf\u4e2a\u539f\u7d20\u90fd\u80fd\u5728\u7ed9\u5b9a\u539f\u70b9\u7684\u524d\u63d0\u4e0b\u5728$\\mathcal{V}$\u4e2d\u627e\u5230\u552f\u4e00\u5bf9\u5e94\uff08\u53cd\u4e4b\u5e76\u4e0d\u5fc5\u4ea6\u7136\uff09\uff0c\u4ece\u800c\u89c4\u5b9a$\\Phi_o$\u4e3a\u5355\u5c04\u975e\u6ee1\u5c04\u3002<\/p>\n\n\n\n<p>\u6211\u4e0d\u786e\u5b9a\u79bb\u5f00\u67d0\u6570\u96c6\uff0c\u5149\u9760\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u7684\u516c\u8bbe\u4f53\u7cfb\u80fd\u5426\u72ec\u7acb\u5b9a\u4e49\u201c\u7ebf\u6bb5\u7684\u957f\u5ea6\u201d\u3001\u201c\u4e24\u7ebf\u6bb5\u957f\u5ea6\u76f8\u7b49\u201d\u3001\u201c\u4e00\u6761\u7ebf\u6bb5\u7684\u957f\u5ea6\u5927\u4e8e\u53e6\u4e00\u6761\u7ebf\u6bb5\u7684\u957f\u5ea6\u201d\u3002\u6211\u4f3c\u61c2\u975e\u61c2\u5730\u770b\u4e86\u4e00\u4e0bWikipedia\uff0c\u5174\u8bb8<a rel=\"noreferrer noopener\" href=\"https:\/\/en.wikipedia.org\/wiki\/Tarski%27s_axioms\" target=\"_blank\">Tarski\u7684\u516c\u8bbe\u4f53\u7cfb<\/a>\u5b9e\u73b0\u4e86\u8fd9\u4ef6\u4e8b\u3002\u4f46\u4e0d\u7ba1\u5982\u4f55\uff0c\u6211\u4eec\u5728\u201c\u957f\u5ea6\u201d\u95ee\u9898\u4e0a\u7684\u91cd\u65b0\u5b9a\u4e49\uff0c\u5e94\u8be5\u4e0d\u4f1a\u9020\u6210\u6574\u4e2a\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u7684\u91cd\u65b0\u5b9a\u4e49\uff0c\u56e0\u4e3a\u201c\u76f8\u7b49\u201d\u3001\u201c\u5927\u5c0f\u201d\u4e4b\u7c7b\u7684\u903b\u8f91\uff0c\u5728\u6570\u96c6\u91cc\u8ddf\u5728\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u91cc\u662f\u76f8\u540c\u7684\u3002\u6240\u4ee5\u4e0b\u4e00\u6b65\u6211\u4eec\u5b9a\u4e49\uff0c$\\mathcal{E}$\u4e2d\u6709\u5411\u7ebf\u6bb5\uff08\u8bb0\u4e3a$\\overline{xy},x,y\\in\\mathcal{E}$\uff09\u7684\u957f\u5ea6\u5c31\u662f\u6240\u5bf9\u5e94\u7684$\\mathcal{V}$\u4e2d\u5411\u91cf\u7684\u8303\uff1a<\/p>\n\n\n\n<p>$$d:\\mathcal{E}\\times\\mathcal{E}\\rightarrow\\mathbb{R},\\overline{ab}\\equiv d\\left(a,b\\right)\\equiv\\left\\|\\Phi\\left(a,b\\right)\\right\\|$$<\/p>\n\n\n\n<p>\u5176\u4e2d$\\left\\|\\mathbf{u}\\right\\|\\equiv\\sqrt{\\mathbf{u}\\cdot\\mathbf{u}},\\mathbf{u}\\in\\mathcal{V}$\u3002<\/p>\n\n\n\n<p>\u540c\u6837\u7684\u9053\u7406\uff0c\u5982\u679c\u80fd\u8ba4\u4e3a\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u539f\u672c\u7684\u201c\u89d2\u5ea6\u5927\u5c0f\u201d\u4ee5\u53ca\u5176\u4e8e\u8fd9\u4e2a\u6982\u5ff5\u7684\u6210\u7acb\u800c\u8bc1\u5f97\u7684\u4e00\u5207\u63a8\u8bba\u4e5f\u662f\u901a\u8fc7\u201c\u76f8\u7b49\u201d\u3001\u201c\u5927\u5c0f\u201d\u7b49\u903b\u8f91\u72ec\u7acb\u4e8e\u6570\u96c6\u800c\u88ab\u89c4\u5b9a\u7684<span id=\"1b261af0-b3f5-4112-a82b-1af9d494a33d\" data-note=\"\u4f8b\u5982\uff0c\u76f4\u89d2\u5728\u300a\u51e0\u4f55\u539f\u672c\u300b\u91cc\uff08\u5b9a\u4e4910\uff09\u5c31\u4e0d\u662f\u901a\u8fc7\u89d2\u5ea6\u5b9a\u4e49\u7684\u3002\u4e00\u4e2a\u666e\u901a\u89d2\u7684\u5927\u5c0f\uff0c\u53c8\u53ef\u4ee5\u6309\u7167\u5176\u6b63\u5207\u7684\u5b9a\u4e49\uff0c\u4ece\u5176\u6240\u5728\u7684\u67d0\u76f4\u89d2\u4e09\u89d2\u5f62\u4e2d\u5bf9\u8fb9\u957f\u5ea6\u4e0e\u90bb\u8fb9\u957f\u5ea6\u4e4b\u6bd4\u6765\u5b9a\u4e49\uff0c\u4ece\u800c\u56de\u5f52\u5230\u5df2\u89e3\u51b3\u4e86\u7684\u7ebf\u6bb5\u957f\u5ea6\u5927\u5c0f\u7684\u95ee\u9898\u3002\" contenteditable=\"false\" class=\"abt-footnote\"><sup>\u200b*\u200b<\/sup><\/span>\uff0c\u90a3\u4e48\u91c7\u7528\u5b9e\u6570\u96c6\u6765\u91cd\u65b0\u89c4\u5b9a\u5c31\u4e0d\u4f1a\u6539\u53d8\u539f\u6709\u903b\u8f91\u57fa\u7840\uff0c\u4ece\u800c\u539f\u6709\u7684\u51e0\u4f55\u63a8\u8bba\u4e5f\u90fd\u6210\u7acb\uff08\u53ef\u7528\uff09\u3002\u73b0\u5728\u6211\u4eec\u901a\u8fc7\u65bd\u74e6\u8328\u4e0d\u7b49\u5f0f\u6765\u5b9a\u4e49$\\mathcal{V}$\u7531\u4e09\u4e2a\u4e0d\u540c\u7684\u70b9\u6240\u6784\u6210\u7684\u89d2\u7684\u5ea6\u6570\u3002<\/p>\n\n\n\n<p>\u5728$\\mathcal{V}$\u4e2d\uff0c\u7531\u5185\u79ef\u8fd0\u7b97\u7684\u6027\u8d28\uff0c\u6709\u5982\u4e0b\u5173\u7cfb\uff1a<\/p>\n\n\n\n<p>\\begin{equation}\\left\\|\\Phi_o\\left(x\\right)\\right\\|^2+\\left\\|\\Phi_o\\left(y\\right)\\right\\|^2=\\left\\|\\Phi_o\\left(x\\right)-\\Phi_o\\left(y\\right)\\right\\|+2\\Phi_o\\left(x\\right)\\cdot\\Phi_o\\left(y\\right)\\label{eq1}\\end{equation}<\/p>\n\n\n\n<p>\u7531\u65bd\u74e6\u8328\u4e0d\u7b49\u5f0f<\/p>\n\n\n\n<p>\\begin{align*}&amp;\\left\\|\\Phi_o\\left(x\\right)\\right\\|^2+\\left\\|\\Phi_o\\left(y\\right)\\right\\|^2\\geq\\left|\\Phi_o\\left(x\\right)\\cdot\\Phi_o\\left(y\\right)\\right|^2\\\\<br \/>\\Leftrightarrow&amp;-1\\leq\\frac{\\Phi_o\\left(x\\right)\\cdot\\Phi_o\\left(y\\right)}{\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|}\\leq1\\end{align*}<\/p>\n\n\n\n<p>\u7531\u4e8e$\\cos:\\left[0,\\pi\\right]\\rightarrow\\left[-1,1\\right]$\u662f\u53cc\u5c04\uff0c\u6545\u53ef\u5b9a\u4e49\u201c\u4e24\u5411\u91cf\u7684\u5939\u89d2\u201d\uff1a<\/p>\n\n\n\n<p>$$\\theta:\\mathcal{V}\\times\\mathcal{V}\\rightarrow\\left[0,\\pi\\right],\\theta\\left(\\mathbf{u},\\mathbf{v}\\right)\\equiv\\mathrm{arccos}\\frac{\\mathbf{u}\\cdot\\mathbf{v}}{\\left\\|\\mathbf{u}\\right\\|\\left\\|\\mathbf{v}\\right\\|}$$<\/p>\n\n\n\n<p>\u5f0f(\\ref{eq1})\u53d8\u4e3a\uff1a<\/p>\n\n\n\n<p>\\begin{align}&amp;\\left\\|\\Phi_o\\left(x\\right)\\right\\|^2+\\left\\|\\Phi_o\\left(y\\right)\\right\\|^2\\\\<br \/>=&amp;\\left\\|\\Phi_o\\left(x\\right)-\\Phi_o\\left(y\\right)\\right\\|^2+2\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|\\cos\\theta\\left(\\Phi_o\\left(x\\right),\\Phi_o\\left(y\\right)\\right)\\label{eq1a}\\end{align}<\/p>\n\n\n\n<p>\u7531\u5185\u79ef\u5b9a\u4e49\u53ef\u77e5$\\theta\\left(\\mathbf{u},\\mathbf{v}\\right)=\\theta\\left(\\mathbf{v},\\mathbf{u}\\right)\\forall\\mathbf{u},\\mathbf{v}\\in\\mathcal{V}$\u3002<\/p>\n\n\n\n<p>\u5728$\\mathcal{E}$\u4e2d\uff0c\u7531\u4f59\u5f26\u5b9a\u7406\uff08\u300a\u51e0\u4f55\u539f\u672c\u300b\u547d\u989812\u300113\uff09\uff0c<\/p>\n\n\n\n<p>\\begin{align}<br \/>&amp;\\overline{ox}^2+\\overline{oy}^2=\\overline{xy}^2+2\\overline{ox}\\times\\overline{oy}\\cos\\angle xoy\\nonumber\\\\<br \/>\\Leftrightarrow&amp;\\left\\|\\Phi_o\\left(x\\right)\\right\\|^2+\\left\\|\\Phi_o\\left(y\\right)\\right\\|^2=\\left\\|\\Phi_x\\left(y\\right)\\right\\|^2+2\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|\\cos\\angle xoy\\label{eq2}\\end{align}<\/p>\n\n\n\n<p>\u5176\u4e2d$\\angle xoy,x,o,y\\in\\mathcal{E}$\u8868\u793a\u7531\u70b9$x$\u3001$o$\u3001$y$\u6784\u6210\u7684\u89d2\u3002\u5c06\u5f0f(\\ref{eq1a})\u4ee3\u5165\u5f0f(\\ref{eq2})\u5f97\uff1a<\/p>\n\n\n\n<p>\\begin{align*}<br \/>&amp;\\left\\|\\Phi_o\\left(x\\right)-\\Phi_o\\left(y\\right)\\right\\|^2+2\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|\\cos\\theta\\left(\\Phi_o\\left(x\\right),\\Phi_o\\left(y\\right)\\right)\\\\<br \/>=&amp;\\left\\|\\Phi_x\\left(y\\right)\\right\\|+2\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|\\cos\\angle xoy\\end{align*}<\/p>\n\n\n\n<p>\u6bd4\u8f83\u4e0a\u5f0f\u7b49\u53f7\u5de6\u53f3\u4e24\u8fb9\uff0c\u53ef\u77e5\u4ee5\u4e0b\u4e24\u547d\u9898\u4e92\u4e3a\u5145\u8981\u6761\u4ef6\uff1a<\/p>\n\n\n\n<p>\\[\\Phi_o\\left(x\\right)-\\Phi_o\\left(y\\right)=\\pm\\Phi_x\\left(y\\right)\\Leftrightarrow\\cos\\angle xoy=\\cos\\theta\\left(\\Phi_o\\left(x\\right),\\Phi_o\\left(y\\right)\\right)\\]<\/p>\n\n\n\n<p>\u6211\u4eec\u4e0d\u59a8\u5bf9\u6620\u5c04$\\Phi$\u589e\u52a0\u89c4\u5b9a\uff1a$\\Phi\\left(o,y\\right)-\\Phi\\left(o,x\\right)=\\Phi\\left(x,y\\right)=-\\Phi\\left(y,x\\right),\\forall o,x,y\\in\\mathcal{E}$\uff0c\u5219\u7531\u4e0a\u8ff0\u7b49\u4ef7\u5173\u7cfb\u6211\u4eec\u540c\u65f6\u83b7\u5f97\u4e86$\\mathcal{E}$\u4e2d\u7684\u89d2$\\angle xoy,x,o,y\\in\\mathcal{E}$\u7684\u5927\u5c0f\u4e0e\u5b9e\u6570\u57df$\\left[0,\\pi\\right]$\u7684\u6620\u5c04\uff1a$\\measuredangle:\\mathcal{E}^3\\rightarrow\\left[0,\\pi\\right],\\measuredangle\\left(xoy\\right)\\equiv\\measuredangle xoy\\equiv\\theta\\left(\\Phi_o\\left(x\\right),\\Phi_o\\left(y\\right)\\right)$\u3002<\/p>\n\n\n\n<p>\u81f3\u6b64\uff0c\u6211\u505a\u7684\u4e8b\u60c5\u662f\u662f\uff1a1\uff09\u89c4\u5b9a\u4e86$\\mathcal{E}$\u4e2d\u7684\u6709\u5411\u7ebf\u6bb5\u4e0e$\\mathcal{V}$\u4e2d\u7684\u5411\u91cf\u7684\u6620\u5c04\uff08\u672a\u89c4\u5b9a\u662f\u4ec0\u4e48\u5c04\uff09\uff1b2\uff09\u5229\u7528$\\mathcal{V}$\u7684\u5185\u79ef\u5b9a\u4e49\uff0c\u89c4\u5b9a\u4e86$\\mathcal{E}$\u4e2d\u7ebf\u6bb5\u7684\u957f\u5ea6\u548c\u89d2\u7684\u5927\u5c0f\u4e0e\u5b9e\u6570\u96c6\u7684\u6620\u5c04\u3002\u5728\u505a\u8fd9\u4e9b\u89c4\u5b9a\u7684\u65f6\u5019\uff0c\u6211\u5229\u7528\u4e86\u5df2\u77e5\u7684\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u63a8\u8bba\uff08\u4f59\u5f26\u5b9a\u7406\uff09\u3002\u8fd9\u4e00\u89c4\u5b9a\u6027\u81ea\u7136\u800c\u7136\u5e26\u6765\u4e00\u4e9b\u9700\u8981\u91cd\u65b0\u8bc1\u660e\u7684\u547d\u9898\u3002\u4f8b\u5982\uff0c\u539f\u51e0\u4f55\u89c4\u5b9a\uff08Tarski\uff09\u7684\u201c\u4e24\u7ebf\u6bb5\u76f8\u7b49\u201d\uff0c\u5219\u5b83\u4eec\u6240\u5bf9\u5e94\u7684\u5b9e\u6570\u503c\u76f8\u7b49\uff1b\u539f\u51e0\u4f55\u5b9a\u4e49\u7684\u76f4\u89d2\uff0c\u5728\u73b0\u6709\u7684\u89c4\u5b9a\u91cc\u5bf9\u5e94\u5b9e\u6570$\\pi\/2$\u7b49\u7b49\u3002\u8fd9\u91cc\u6211\u672a\u80fd\u5b8c\u5907\u5730\u5217\u51fa\u4e0a\u8ff0\u91cd\u65b0\u89c4\u5b9a\u6240\u5f15\u8d77\u7684\u8fd9\u7c7b\u91cd\u65b0\u8bc1\u660e\u4efb\u52a1\uff0c\u4f46\u6211\u76f8\u4fe1\u8fd9\u4e9b\u547d\u9898\u662f\u6709\u9650\u7684\uff0c\u6613\u8bc1\u7684\u3002\u8fd9\u4e3a\u6211\u4eec\u5efa\u7acb$\\mathcal{E}$\u4e0e$\\mathcal{V}$\u7684\u7ef4\u6570\u5bf9\u5e94\u6027\u6253\u4e0b\u4e86\u90e8\u5206\u57fa\u7840\uff0c\u5373$\\mathcal{E}$\u7684\u201c\u76f4\u89d2\u201d\u4e0e$\\mathcal{V}$\u7684\u201c\u6b63\u4ea4\u5411\u91cf\u201d\u5bf9\u5e94\u4e0a\u4e86\u3002<\/p>\n\n\n\n<p>\u5c1a\u7f3a\u5c11\u7684\u6982\u5ff5\u5bf9\u5e94\u662f\u201c\u8fc7\u4e00\u70b9\u7684\u76f4\u7ebf\u201d\u3002\u4e5f\u6b63\u662f\u5728\u8fd9\u4e2a\u95ee\u9898\u91cc\uff0c\u6211\u53d1\u73b0$\\Phi_o$\u4e0d\u5f97\u4e0d\u5982\u73b0\u6709\u7684\u4e66\u4e2d\u90a3\u822c\u8fdb\u4e00\u6b65\u89c4\u5b9a\u4e3a\u53cc\u5c04\u3002\u5148\u770b\u4e00\u4e2a\u51fa\u53d1\u70b9\u6bd4\u8f83\u597d\uff0c\u4f46\u5931\u8d25\u7684\u4f8b\u5b50\u2014\u2014\u4ece$\\mathcal{E}$\u7684\u5df2\u77e5\u6982\u5ff5\u51fa\u53d1\uff0c\u901a\u8fc7\u5df2\u89c4\u5b9a\u7684\uff08\u672a\u5fc5\u53cc\u5c04\u7684\uff09\u6620\u5c04$\\Phi_o$\u6765\u627e\u5230$\\mathcal{V}$\u4e2d\u7684\u5bf9\u5e94\u3002<\/p>\n\n\n\n<p> \u7531\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u6613\u77e5\uff0c\u8fc7\u70b9$a$\u3001$b$\u7684\u76f4\u7ebf\u4e0a\u7684\u4efb\u4e00\u70b9$c$\uff0c$a,b,c\\in\\mathcal{E},a\\neq b$\uff0c\u5fc5\u6ee1\u8db3\u4ee5\u4e0b\u56db\u79cd\u60c5\u51b5\u4e4b\u4e00\uff1a1\uff09$\\measuredangle cab=0$\uff1b2\uff09$\\measuredangle cab=\\pi$\uff1b3\uff09$c=a$\uff1b4\uff09$c=b$\u3002\u6839\u636e\u524d\u6587\u7684\u89c4\u5b9a\uff0c\u53c8\u6709$\\overline{ac}=\\left\\|\\Phi_a\\left(c\\right)\\right\\|$\u3001$\\overline{ab}=\\left\\|\\Phi_a\\left(b\\right)\\right\\|$\u3002\u5bf9\u4e8e\u60c5\u51b51\uff09\u548c2\uff09\uff0c\u7531<\/p>\n\n\n\n<p>$$\\cos\\measuredangle cab =\\frac{ \\Phi_a\\left(c\\right)\\cdot\\Phi_a\\left(b\\right)}{\\left\\|\\Phi_a\\left(c\\right)\\right\\|\\left\\|\\Phi_a\\left(b\\right)\\right\\|}=\\pm 1$$<\/p>\n\n\n\n<p>\u5de6\u53f3\u4e24\u8fb9\u540c\u9664$\\left\\|\\Phi_a\\left(b\\right)\\right\\|$\u6709<\/p>\n\n\n\n<p>$$\\frac{\\Phi_a\\left(c\\right)\\cdot\\Phi_a\\left(b\\right)}{\\Phi_a\\left(b\\right)\\cdot\\Phi_a\\left(b\\right)}=\\pm\\frac{\\left\\|\\Phi_a\\left(c\\right)\\right\\|}{\\left\\|\\Phi_a\\left(b\\right)\\right\\|}\\equiv\\alpha^\\prime$$<\/p>\n\n\n\n<p>\u5176\u4e2d\u60c5\u51b51\uff09\u53d6\u6b63\u53f7\uff0c\u60c5\u51b52\uff09\u53d6\u8d1f\u53f7\u3002\u53e6\u5916\uff0c\u663e\u7136\u60c5\u51b53\uff09\u548c4\uff09$\\alpha^\\prime$\u7b49\u4e8e0\u6216\u65e0\u7a77\u3002\u6211\u4eec\u53ef\u4ee5\u91cd\u65b0\u6574\u7406\u56db\u79cd\u60c5\u51b5\uff0c\u5199\u6210\\begin{align*}&amp;\\Phi_a\\left(c\\right)\\cdot\\Phi_a\\left(b\\right)=\\alpha\\left(\\Phi_a\\left(b\\right)\\cdot\\Phi_a\\left(b\\right)\\right),\\alpha\\in\\mathbb{R}\\\\\\Leftarrow&amp;\\Phi_a\\left(c\\right)=\\alpha\\Phi_a\\left(b\\right)\\end{align*}<\/p>\n\n\n\n<p>\u4e0a\u5f0f\u4e2d\uff0c\u524d\u8005\u4ec5\u662f\u540e\u8005\u7684\u5fc5\u8981\u975e\u5145\u4efd\u6761\u4ef6\u3002\u5982\u679c\u8fd9\u4e00\u6b65\u662f\u5145\u8981\u6761\u4ef6\uff0c\u4ece\u800c\u5f97\u5230\u6700\u540e\u7684\u7ed3\u8bba\uff0c\u90a3\u4e48\u6211\u4eec\u65e0\u9700\u89c4\u5b9a$\\Phi_a$\u662f\u53cc\u5c04\uff0c\u5c31\u80fd\u628a$\\mathcal{E}$\u4e2d\u8fc7\u4efb\u610f\u4e24\u70b9$a,b$\u7684\u76f4\u7ebf\u4e0a\u7684\u70b9\u7684\u96c6\u5408\uff0c\u6620\u5c04\u4e3a$\\mathcal{V}$\u7684\u5b50\u96c6$\\mathcal{L}_{ab}=\\left\\{\\mathbf{c}|\\Phi_a\\left(c\\right)=\\alpha\\Phi_a\\left(b\\right),\\alpha\\in\\mathbb{R}\\}$\u3002\u4e8b\u5b9e\u4e0a\uff0c\u6613\u8bc1\u8fd9\u4e00\u5b50\u96c6\u662f$\\mathcal{V}$\u7684\u5b50\u7a7a\u95f4\u4e14\u7ef4\u6570\u662f1\u3002\u82e5\u6709\u4e86\u6b64\u57fa\u7840\uff0c\u5c31\u53ef\u8fdb\u4e00\u6b65\u63a8\u6f14\u51fa$\\mathcal{E}$\u7684\u7ef4\u6570\u5c31\u662f$\\mathcal{V}$\u7684\u7ef4\u6570\uff0c\u4ece\u5934\u5230\u5c3e\u65e0\u9700\u8bf4\u660e$\\Phi_a$\u662f\u53cc\u5c04\u3002\u53e6\u5916\uff0c\u8fd9\u4e5f\u540c\u65f6\u8bc1\u660e\u4e86\u53e6\u4e00\u4e2a\u91cd\u8981\u7684\u7ed3\u679c\uff0c\u5c31\u662f\u628a$\\mathcal{E}$\u4e2d\u7684\u76f4\u7ebf\u4e0e\u5b9e\u6570\u8f74\u5efa\u7acb\u4e86\u5bf9\u5e94\uff0c\u4e3a\u5efa\u7acb\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u5960\u5b9a\u4e86\u57fa\u7840\u3002\u53ef\u60dc\uff0c\u5728\u4e0a\u9762\u6211\u4eec\u53ea\u80fd\u5f97\u5230\u5fc5\u8981\u975e\u5145\u4efd\u6761\u4ef6\u7684\u5173\u7cfb\uff0c\u6240\u4ee5\u6211\u4eec\u5c31\u65e0\u6cd5\u62e5\u6709\u540e\u9762\u7684\u8fd9\u4e9b\u4f18\u7f8e\u53d9\u8ff0\u3002<\/p>\n\n\n\n<p>\u5982\u679c\u6211\u4eec\u53cd\u8fc7\u6765\uff0c\u4ece$\\mathcal{V}$\u51fa\u53d1\u53bb\u627e$\\mathcal{E}$\u4e2d\u7684\u76f4\u7ebf\uff0c\u5c31\u514d\u4e0d\u4e86\u201c\u7ed9\u5b9a\u4efb\u4e00$\\mathbb{u}\\in\\mathcal{V}$\uff0c\u627e\u51fa\u5176\u5728$\\mathcal{E}$\u4e2d\u5bf9\u5e94\u7684\u6709\u5411\u7ebf\u6bb5\u201d\u7684\u4efb\u52a1\uff0c\u8fd9\u8981\u6c42\u6620\u5c04$\\Phi_a$\u662f\u53ef\u9006\u7684\uff0c\u5373\u4e3a\u53cc\u5c04\u3002\u867d\u7136\u6211\u770b\u4e0d\u5230$\\Phi_a$\u6709\u4ec0\u4e48\u7406\u7531\u4e0d\u80fd\u662f\u4e00\u4e2a\u53cc\u5c04\uff0c\u4f46\u6bd5\u7adf\u8fd9\u662f\u4e00\u9879\u91cd\u5927\u7684\u201c\u4ece$\\mathcal{V}$\u5230$\\mathcal{E}$\u7684\u53cd\u5411\u89c4\u5b9a\u201d\u52a8\u4f5c\uff0c\u8fd9\u6bd4\u524d\u9762\u7684\u201c\u957f\u5ea6\u201d\u548c\u201c\u89d2\u5ea6\u201d\u7684\u53cd\u5411\u89c4\u5b9a\u5f71\u54cd\u5927\u591a\u4e86\uff0c\u662f\u5168\u5c40\u6027\u5730\u89c4\u5b9a\u4e86$\\mathcal{E}$\u8fd9\u6574\u4e2a\u7a7a\u95f4\u3002\u5982\u6b64\u4e00\u6765\uff0c$\\mathcal{E}$\u4e0d\u4ec5\u5e26\u7740\u5176\u539f\u6709\u7684\u300a\u51e0\u4f55\u539f\u672c\u300b\u6240\u6709\u63a8\u8bba\uff0c\u53c8\u7a81\u7136\u4e8e\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u7b49\u8ddd\u540c\u6784\u3002\u51ed\u4ec0\u4e48\u5411\u91cf\u7a7a\u95f4\u2014\u2014\u4e00\u4e2a\u610f\u56fe\u5b8c\u5168\u4e0e\u51e0\u4f55\u65e0\u5173\u7684\u6570\u5b66\u6982\u5ff5\u2014\u2014\u6070\u597d\u91cd\u73b0\u4e86\u300a\u51e0\u4f55\u539f\u672c\u300b\u7684\u6b27\u51e0\u91cc\u5fb7\u7a7a\u95f4\u7684\u4e00\u5207\u6027\u8d28\u5462\uff1f\u6211\u627e\u4e0d\u5230\u5728\u8fd9\u4e2a\u5c42\u9762\u4e0a\u6709\u4ec0\u4e48\u4e07\u4f5b\u671d\u5b97\u5f0f\u7684\u89e3\u91ca\u3002\u5982\u679c\u6ca1\u6709\u8fd9\u79cd\u89e3\u91ca\uff0c\u90a3\u4e48\u5c31\u53ea\u80fd\u8ba4\u4e3a\uff0c\u8fd9\u4e00\u53cc\u5c04\u53ea\u662f\u91cd\u65b0\u5b9a\u4e49\u4e86\u4e00\u4e2a\u7a7a\u95f4$\\mathcal{E}$\uff0c\u5176\u51e0\u4f55\u6027\u8d28\u53ea\u7531\u5b9a\u4e49\u5b83\u7684\u90a3\u4e2a\u5185\u79ef\u7a7a\u95f4$\\mathcal{V}$\u6765\u89c4\u5b9a\u2014\u2014\u8fd9\u4e5f\u662f\u73b0\u4ee3\u51e0\u4f55\u5bf9\u201c\u6b27\u51e0\u91cc\u5fb7\u7a7a\u95f4\u201d\u7684\u5b9a\u4e49\uff0c\u5373\u5b8c\u5168\u629b\u5f03\u4e86\u300a\u51e0\u4f55\u539f\u672c\u300b\u3002\u65e2\u7136\u5982\u6b64\uff0c\u8fd9\u4e00\u73b0\u4ee3\u5b9a\u4e49\u7684\u201c\u6b27\u51e0\u91cc\u5fb7\u7a7a\u95f4\u201d\u4e2d\uff0c\u662f\u5426\u8981\u7528\u5411\u91cf\u4ee3\u6570\u91cd\u65b0\u8bc1\u660e\u4e00\u5207\u6570\u5b66\u53f2\u4e0a\u4ece\u300a\u51e0\u4f55\u539f\u672c\u300b\u8bc1\u660e\u8fc7\u7684\u51e0\u4f55\u5b9a\u7406\uff1f\u8fd9\u4e2a\u4efb\u52a1\u662f\u4e00\u4e2a\u5e9e\u5927\u7684\u4efb\u52a1\uff0c\u8fd8\u662f\u53ea\u9700\u91cd\u73b0\u51fa\u6b27\u51e0\u7406\u5fb7\u51e0\u4f55\u7684\u6709\u9650\u51e0\u6761\u516c\u8bbe\u5373\u53ef\uff1f\u8fd9\u4e2a\u4efb\u52a1\u662f\u5426\u5df2\u7ecf\u88ab\u5b8c\u6210\u4e86\uff1f\u5c3d\u7ba1\u5982\u6b64\uff0c\u6211\u4eec\u96be\u9053\u771f\u7684\u4e0d\u80fdrelax\u6389\u201c\u53cc\u5c04\u201d\u8fd9\u4e00\u89c4\u5b9a\u6027\u5417\uff1f<\/p>\n\n\n\n<section aria-label=\"Footnotes\" class=\"wp-block-abt-footnotes abt-footnotes\" role=\"region\"><hr \/><ol><li class=\"abt-footnotes-item\" id=\"1b261af0-b3f5-4112-a82b-1af9d494a33d-ref\"><div class=\"abt-footnotes-item__marker\"><sup>\u200b*\u200b<\/sup><\/div><div class=\"abt-footnotes-item__content\">\u4f8b\u5982\uff0c\u76f4\u89d2\u5728\u300a\u51e0\u4f55\u539f\u672c\u300b\u91cc\uff08\u5b9a\u4e4910\uff09\u5c31\u4e0d\u662f\u901a\u8fc7\u89d2\u5ea6\u5b9a\u4e49\u7684\u3002\u4e00\u4e2a\u666e\u901a\u89d2\u7684\u5927\u5c0f\uff0c\u53c8\u53ef\u4ee5\u6309\u7167\u5176\u6b63\u5207\u7684\u5b9a\u4e49\uff0c\u4ece\u5176\u6240\u5728\u7684\u67d0\u76f4\u89d2\u4e09\u89d2\u5f62\u4e2d\u5bf9\u8fb9\u957f\u5ea6\u4e0e\u90bb\u8fb9\u957f\u5ea6\u4e4b\u6bd4\u6765\u5b9a\u4e49\uff0c\u4ece\u800c\u56de\u5f52\u5230\u5df2\u89e3\u51b3\u4e86\u7684\u7ebf\u6bb5\u957f\u5ea6\u5927\u5c0f\u7684\u95ee\u9898\u3002<\/div><\/li><\/ol><\/section>\n\n\n\n<p><\/p>\n\n\n\n<section aria-label=\"Bibliography\" class=\"wp-block-abt-bibliography abt-bibliography\" role=\"region\"><ol class=\"abt-bibliography__body\" data-entryspacing=\"1\" data-hangingindent=\"true\" data-linespacing=\"1\"><li id=\"1278065742\">  <div class=\"csl-entry\">Audin M (2002) Geometry. Springer<\/div>\n<\/li><li id=\"1414259973\">  <div class=\"csl-entry\">Berger M (1987) Geometry I. Springer<\/div>\n<\/li><\/ol><\/section>\n<!-- kcite active, but no citations found -->\n<\/div> <!-- kcite-section 8007 -->","protected":false},"excerpt":{"rendered":"<p>[latexpage] $\\mathcal{E}$\u662f\u4e00\u4e2a\u975e\u7a7a\u96c6\u5408\uff0c\u91cc\u9762\u7684\u5143\u7d20\u79f0\u4e3a\u201c\u70b9\u201d\u3002\u5bf9\u4e8e\u96c6\u5408\u4e2d\u7684\u8fd9\u4e9b\u70b9\uff0c\u6211\u4eec\u53ef\u4ee5\u505a\u300a\u51e0\u4f55\u539f\u672c\u300b\u8bf4\u80fd\u505a\u7684\u6240\u6709\u4e8b\uff0c\u4ece\u800c\u5f97\u5230\u4e00\u5207\u300a\u51e0\u4f55\u539f\u672c\u300b\u80fd\u5f97\u5230\u7684\u7ed3\u8bba\u3002\u5176\u4e2d\u5c31\u5305\u62ec\u5141\u8bb8\u6211\u4eec\u8c08\u53ca\u201c\u7531\u70b9$a$\u5230\u70b9$b$\u7684\u6709\u5411\u7ebf\u6bb5\u201d\uff0c$a,b\\in\\mathcal{E}$\u3002\u6211\u4eec\u5efa\u7acb\u4e00\u4e2a\u201c\u4ece\u70b9$a$\u5230\u70b9$b$\u7684\u6709\u5411\u7ebf\u6bb5\u201d\u5230\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4$\\mathcal{V}$\u7684\u6620\u5c04$\\Phi:\\mathcal{E}\\times\\mathcal{E}\\rightarrow\\mathcal{V}$\uff0c\u4e14$\\Phi\\left(x,x\\right)\\equiv\\mathbf{0}\\forall x\\in\\mathcal{E}$\uff0c\u5176\u4e2d$\\mathbf{0}$\u662f$\\mathcal{V}$\u7684\u96f6\u5411\u91cf\u3002\u9009\u5b9a\u4e00\u4e2a\u70b9$o\\in\\mathcal{E}$\uff0c\u53c8\u53ef\u8bb0$\\Phi_o:\\mathcal{E}\\rightarrow\\mathcal{V}$\uff0c$\\Phi_o\\left(x\\right)\\equiv\\Phi\\left(o,x\\right)$\u3002\u5728\u5f88\u591a\u4e66\u4e2d\u200b(Berger 1987; Audin 2002)\u200b\uff0c\u76f4\u63a5\u9884\u5148\u628a$\\Phi_o$\u89c4\u5b9a\u4e3a\u53cc\u5c04\u3002\u4f46\u6211\u5728\u8fd9\u91cc\u60f3\u5148\u4e0d\u4f5c\u6b64\u89c4\u5b9a\u3002\u6700\u591a\uff0c\u7531\u4e8e\u6211\u4eec\u5e0c\u671b$\\mathcal{E}$\u4e2d\u7684\u6bcf\u4e2a\u539f\u7d20\u90fd\u80fd\u5728\u7ed9\u5b9a\u539f\u70b9\u7684\u524d\u63d0\u4e0b\u5728$\\mathcal{V}$\u4e2d\u627e\u5230\u552f\u4e00\u5bf9\u5e94\uff08\u53cd\u4e4b\u5e76\u4e0d\u5fc5\u4ea6\u7136\uff09\uff0c\u4ece\u800c\u89c4\u5b9a$\\Phi_o$\u4e3a\u5355\u5c04\u975e\u6ee1\u5c04\u3002 \u6211\u4e0d\u786e\u5b9a\u79bb\u5f00\u67d0\u6570\u96c6\uff0c\u5149\u9760\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u7684\u516c\u8bbe\u4f53\u7cfb\u80fd\u5426\u72ec\u7acb\u5b9a\u4e49\u201c\u7ebf\u6bb5\u7684\u957f\u5ea6\u201d\u3001\u201c\u4e24\u7ebf\u6bb5\u957f\u5ea6\u76f8\u7b49\u201d\u3001\u201c\u4e00\u6761\u7ebf\u6bb5\u7684\u957f\u5ea6\u5927\u4e8e\u53e6\u4e00\u6761\u7ebf\u6bb5\u7684\u957f\u5ea6\u201d\u3002\u6211\u4f3c\u61c2\u975e\u61c2\u5730\u770b\u4e86\u4e00\u4e0bWikipedia\uff0c\u5174\u8bb8Tarski\u7684\u516c\u8bbe\u4f53\u7cfb\u5b9e\u73b0\u4e86\u8fd9\u4ef6\u4e8b\u3002\u4f46\u4e0d\u7ba1\u5982\u4f55\uff0c\u6211\u4eec\u5728\u201c\u957f\u5ea6\u201d\u95ee\u9898\u4e0a\u7684\u91cd\u65b0\u5b9a\u4e49\uff0c\u5e94\u8be5\u4e0d\u4f1a\u9020\u6210\u6574\u4e2a\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u7684\u91cd\u65b0\u5b9a\u4e49\uff0c\u56e0\u4e3a\u201c\u76f8\u7b49\u201d\u3001\u201c\u5927\u5c0f\u201d\u4e4b\u7c7b\u7684\u903b\u8f91\uff0c\u5728\u6570\u96c6\u91cc\u8ddf\u5728\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u91cc\u662f\u76f8\u540c\u7684\u3002\u6240\u4ee5\u4e0b\u4e00\u6b65\u6211\u4eec\u5b9a\u4e49\uff0c$\\mathcal{E}$\u4e2d\u6709\u5411\u7ebf\u6bb5\uff08\u8bb0\u4e3a$\\overline{xy},x,y\\in\\mathcal{E}$\uff09\u7684\u957f\u5ea6\u5c31\u662f\u6240\u5bf9\u5e94\u7684$\\mathcal{V}$\u4e2d\u5411\u91cf\u7684\u8303\uff1a $$d:\\mathcal{E}\\times\\mathcal{E}\\rightarrow\\mathbb{R},\\overline{ab}\\equiv d\\left(a,b\\right)\\equiv\\left\\|\\Phi\\left(a,b\\right)\\right\\|$$ \u5176\u4e2d$\\left\\|\\mathbf{u}\\right\\|\\equiv\\sqrt{\\mathbf{u}\\cdot\\mathbf{u}},\\mathbf{u}\\in\\mathcal{V}$\u3002 \u540c\u6837\u7684\u9053\u7406\uff0c\u5982\u679c\u80fd\u8ba4\u4e3a\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u539f\u672c\u7684\u201c\u89d2\u5ea6\u5927\u5c0f\u201d\u4ee5\u53ca\u5176\u4e8e\u8fd9\u4e2a\u6982\u5ff5\u7684\u6210\u7acb\u800c\u8bc1\u5f97\u7684\u4e00\u5207\u63a8\u8bba\u4e5f\u662f\u901a\u8fc7\u201c\u76f8\u7b49\u201d\u3001\u201c\u5927\u5c0f\u201d\u7b49\u903b\u8f91\u72ec\u7acb\u4e8e\u6570\u96c6\u800c\u88ab\u89c4\u5b9a\u7684\u200b*\u200b\uff0c\u90a3\u4e48\u91c7\u7528\u5b9e\u6570\u96c6\u6765\u91cd\u65b0\u89c4\u5b9a\u5c31\u4e0d\u4f1a\u6539\u53d8\u539f\u6709\u903b\u8f91\u57fa\u7840\uff0c\u4ece\u800c\u539f\u6709\u7684\u51e0\u4f55\u63a8\u8bba\u4e5f\u90fd\u6210\u7acb\uff08\u53ef\u7528\uff09\u3002\u73b0\u5728\u6211\u4eec\u901a\u8fc7\u65bd\u74e6\u8328\u4e0d\u7b49\u5f0f\u6765\u5b9a\u4e49$\\mathcal{V}$\u7531\u4e09\u4e2a\u4e0d\u540c\u7684\u70b9\u6240\u6784\u6210\u7684\u89d2\u7684\u5ea6\u6570\u3002 \u5728$\\mathcal{V}$\u4e2d\uff0c\u7531\u5185\u79ef\u8fd0\u7b97\u7684\u6027\u8d28\uff0c\u6709\u5982\u4e0b\u5173\u7cfb\uff1a \\begin{equation}\\left\\|\\Phi_o\\left(x\\right)\\right\\|^2+\\left\\|\\Phi_o\\left(y\\right)\\right\\|^2=\\left\\|\\Phi_o\\left(x\\right)-\\Phi_o\\left(y\\right)\\right\\|+2\\Phi_o\\left(x\\right)\\cdot\\Phi_o\\left(y\\right)\\label{eq1}\\end{equation} \u7531\u65bd\u74e6\u8328\u4e0d\u7b49\u5f0f \\begin{align*}&amp;\\left\\|\\Phi_o\\left(x\\right)\\right\\|^2+\\left\\|\\Phi_o\\left(y\\right)\\right\\|^2\\geq\\left|\\Phi_o\\left(x\\right)\\cdot\\Phi_o\\left(y\\right)\\right|^2\\\\\\Leftrightarrow&amp;-1\\leq\\frac{\\Phi_o\\left(x\\right)\\cdot\\Phi_o\\left(y\\right)}{\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|}\\leq1\\end{align*} \u7531\u4e8e$\\cos:\\left[0,\\pi\\right]\\rightarrow\\left[-1,1\\right]$\u662f\u53cc\u5c04\uff0c\u6545\u53ef\u5b9a\u4e49\u201c\u4e24\u5411\u91cf\u7684\u5939\u89d2\u201d\uff1a $$\\theta:\\mathcal{V}\\times\\mathcal{V}\\rightarrow\\left[0,\\pi\\right],\\theta\\left(\\mathbf{u},\\mathbf{v}\\right)\\equiv\\mathrm{arccos}\\frac{\\mathbf{u}\\cdot\\mathbf{v}}{\\left\\|\\mathbf{u}\\right\\|\\left\\|\\mathbf{v}\\right\\|}$$ \u5f0f(\\ref{eq1})\u53d8\u4e3a\uff1a \\begin{align}&amp;\\left\\|\\Phi_o\\left(x\\right)\\right\\|^2+\\left\\|\\Phi_o\\left(y\\right)\\right\\|^2\\\\=&amp;\\left\\|\\Phi_o\\left(x\\right)-\\Phi_o\\left(y\\right)\\right\\|^2+2\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|\\cos\\theta\\left(\\Phi_o\\left(x\\right),\\Phi_o\\left(y\\right)\\right)\\label{eq1a}\\end{align} \u7531\u5185\u79ef\u5b9a\u4e49\u53ef\u77e5$\\theta\\left(\\mathbf{u},\\mathbf{v}\\right)=\\theta\\left(\\mathbf{v},\\mathbf{u}\\right)\\forall\\mathbf{u},\\mathbf{v}\\in\\mathcal{V}$\u3002 \u5728$\\mathcal{E}$\u4e2d\uff0c\u7531\u4f59\u5f26\u5b9a\u7406\uff08\u300a\u51e0\u4f55\u539f\u672c\u300b\u547d\u989812\u300113\uff09\uff0c \\begin{align}&amp;\\overline{ox}^2+\\overline{oy}^2=\\overline{xy}^2+2\\overline{ox}\\times\\overline{oy}\\cos\\angle xoy\\nonumber\\\\\\Leftrightarrow&amp;\\left\\|\\Phi_o\\left(x\\right)\\right\\|^2+\\left\\|\\Phi_o\\left(y\\right)\\right\\|^2=\\left\\|\\Phi_x\\left(y\\right)\\right\\|^2+2\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|\\cos\\angle xoy\\label{eq2}\\end{align} \u5176\u4e2d$\\angle xoy,x,o,y\\in\\mathcal{E}$\u8868\u793a\u7531\u70b9$x$\u3001$o$\u3001$y$\u6784\u6210\u7684\u89d2\u3002\u5c06\u5f0f(\\ref{eq1a})\u4ee3\u5165\u5f0f(\\ref{eq2})\u5f97\uff1a \\begin{align*}&amp;\\left\\|\\Phi_o\\left(x\\right)-\\Phi_o\\left(y\\right)\\right\\|^2+2\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|\\cos\\theta\\left(\\Phi_o\\left(x\\right),\\Phi_o\\left(y\\right)\\right)\\\\=&amp;\\left\\|\\Phi_x\\left(y\\right)\\right\\|+2\\left\\|\\Phi_o\\left(x\\right)\\right\\|\\left\\|\\Phi_o\\left(y\\right)\\right\\|\\cos\\angle xoy\\end{align*} \u6bd4\u8f83\u4e0a\u5f0f\u7b49\u53f7\u5de6\u53f3\u4e24\u8fb9\uff0c\u53ef\u77e5\u4ee5\u4e0b\u4e24\u547d\u9898\u4e92\u4e3a\u5145\u8981\u6761\u4ef6\uff1a \\[\\Phi_o\\left(x\\right)-\\Phi_o\\left(y\\right)=\\pm\\Phi_x\\left(y\\right)\\Leftrightarrow\\cos\\angle xoy=\\cos\\theta\\left(\\Phi_o\\left(x\\right),\\Phi_o\\left(y\\right)\\right)\\] \u6211\u4eec\u4e0d\u59a8\u5bf9\u6620\u5c04$\\Phi$\u589e\u52a0\u89c4\u5b9a\uff1a$\\Phi\\left(o,y\\right)-\\Phi\\left(o,x\\right)=\\Phi\\left(x,y\\right)=-\\Phi\\left(y,x\\right),\\forall o,x,y\\in\\mathcal{E}$\uff0c\u5219\u7531\u4e0a\u8ff0\u7b49\u4ef7\u5173\u7cfb\u6211\u4eec\u540c\u65f6\u83b7\u5f97\u4e86$\\mathcal{E}$\u4e2d\u7684\u89d2$\\angle xoy,x,o,y\\in\\mathcal{E}$\u7684\u5927\u5c0f\u4e0e\u5b9e\u6570\u57df$\\left[0,\\pi\\right]$\u7684\u6620\u5c04\uff1a$\\measuredangle:\\mathcal{E}^3\\rightarrow\\left[0,\\pi\\right],\\measuredangle\\left(xoy\\right)\\equiv\\measuredangle xoy\\equiv\\theta\\left(\\Phi_o\\left(x\\right),\\Phi_o\\left(y\\right)\\right)$\u3002 \u81f3\u6b64\uff0c\u6211\u505a\u7684\u4e8b\u60c5\u662f\u662f\uff1a1\uff09\u89c4\u5b9a\u4e86$\\mathcal{E}$\u4e2d\u7684\u6709\u5411\u7ebf\u6bb5\u4e0e$\\mathcal{V}$\u4e2d\u7684\u5411\u91cf\u7684\u6620\u5c04\uff08\u672a\u89c4\u5b9a\u662f\u4ec0\u4e48\u5c04\uff09\uff1b2\uff09\u5229\u7528$\\mathcal{V}$\u7684\u5185\u79ef\u5b9a\u4e49\uff0c\u89c4\u5b9a\u4e86$\\mathcal{E}$\u4e2d\u7ebf\u6bb5\u7684\u957f\u5ea6\u548c\u89d2\u7684\u5927\u5c0f\u4e0e\u5b9e\u6570\u96c6\u7684\u6620\u5c04\u3002\u5728\u505a\u8fd9\u4e9b\u89c4\u5b9a\u7684\u65f6\u5019\uff0c\u6211\u5229\u7528\u4e86\u5df2\u77e5\u7684\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u63a8\u8bba\uff08\u4f59\u5f26\u5b9a\u7406\uff09\u3002\u8fd9\u4e00\u89c4\u5b9a\u6027\u81ea\u7136\u800c\u7136\u5e26\u6765\u4e00\u4e9b\u9700\u8981\u91cd\u65b0\u8bc1\u660e\u7684\u547d\u9898\u3002\u4f8b\u5982\uff0c\u539f\u51e0\u4f55\u89c4\u5b9a\uff08Tarski\uff09\u7684\u201c\u4e24\u7ebf\u6bb5\u76f8\u7b49\u201d\uff0c\u5219\u5b83\u4eec\u6240\u5bf9\u5e94\u7684\u5b9e\u6570\u503c\u76f8\u7b49\uff1b\u539f\u51e0\u4f55\u5b9a\u4e49\u7684\u76f4\u89d2\uff0c\u5728\u73b0\u6709\u7684\u89c4\u5b9a\u91cc\u5bf9\u5e94\u5b9e\u6570$\\pi\/2$\u7b49\u7b49\u3002\u8fd9\u91cc\u6211\u672a\u80fd\u5b8c\u5907\u5730\u5217\u51fa\u4e0a\u8ff0\u91cd\u65b0\u89c4\u5b9a\u6240\u5f15\u8d77\u7684\u8fd9\u7c7b\u91cd\u65b0\u8bc1\u660e\u4efb\u52a1\uff0c\u4f46\u6211\u76f8\u4fe1\u8fd9\u4e9b\u547d\u9898\u662f\u6709\u9650\u7684\uff0c\u6613\u8bc1\u7684\u3002\u8fd9\u4e3a\u6211\u4eec\u5efa\u7acb$\\mathcal{E}$\u4e0e$\\mathcal{V}$\u7684\u7ef4\u6570\u5bf9\u5e94\u6027\u6253\u4e0b\u4e86\u90e8\u5206\u57fa\u7840\uff0c\u5373$\\mathcal{E}$\u7684\u201c\u76f4\u89d2\u201d\u4e0e$\\mathcal{V}$\u7684\u201c\u6b63\u4ea4\u5411\u91cf\u201d\u5bf9\u5e94\u4e0a\u4e86\u3002 \u5c1a\u7f3a\u5c11\u7684\u6982\u5ff5\u5bf9\u5e94\u662f\u201c\u8fc7\u4e00\u70b9\u7684\u76f4\u7ebf\u201d\u3002\u4e5f\u6b63\u662f\u5728\u8fd9\u4e2a\u95ee\u9898\u91cc\uff0c\u6211\u53d1\u73b0$\\Phi_o$\u4e0d\u5f97\u4e0d\u5982\u73b0\u6709\u7684\u4e66\u4e2d\u90a3\u822c\u8fdb\u4e00\u6b65\u89c4\u5b9a\u4e3a\u53cc\u5c04\u3002\u5148\u770b\u4e00\u4e2a\u51fa\u53d1\u70b9\u6bd4\u8f83\u597d\uff0c\u4f46\u5931\u8d25\u7684\u4f8b\u5b50\u2014\u2014\u4ece$\\mathcal{E}$\u7684\u5df2\u77e5\u6982\u5ff5\u51fa\u53d1\uff0c\u901a\u8fc7\u5df2\u89c4\u5b9a\u7684\uff08\u672a\u5fc5\u53cc\u5c04\u7684\uff09\u6620\u5c04$\\Phi_o$\u6765\u627e\u5230$\\mathcal{V}$\u4e2d\u7684\u5bf9\u5e94\u3002 \u7531\u6b27\u51e0\u91cc\u5fb7\u51e0\u4f55\u6613\u77e5\uff0c\u8fc7\u70b9$a$\u3001$b$\u7684\u76f4\u7ebf\u4e0a\u7684\u4efb\u4e00\u70b9$c$\uff0c$a,b,c\\in\\mathcal{E},a\\neq b$\uff0c\u5fc5\u6ee1\u8db3\u4ee5\u4e0b\u56db\u79cd\u60c5\u51b5\u4e4b\u4e00\uff1a1\uff09$\\measuredangle cab=0$\uff1b2\uff09$\\measuredangle cab=\\pi$\uff1b3\uff09$c=a$\uff1b4\uff09$c=b$\u3002\u6839\u636e\u524d\u6587\u7684\u89c4\u5b9a\uff0c\u53c8\u6709$\\overline{ac}=\\left\\|\\Phi_a\\left(c\\right)\\right\\|$\u3001$\\overline{ab}=\\left\\|\\Phi_a\\left(b\\right)\\right\\|$\u3002\u5bf9\u4e8e\u60c5\u51b51\uff09\u548c2\uff09\uff0c\u7531 $$\\cos\\measuredangle cab =\\frac{ \\Phi_a\\left(c\\right)\\cdot\\Phi_a\\left(b\\right)}{\\left\\|\\Phi_a\\left(c\\right)\\right\\|\\left\\|\\Phi_a\\left(b\\right)\\right\\|}=\\pm 1$$ \u5de6\u53f3\u4e24\u8fb9\u540c\u9664$\\left\\|\\Phi_a\\left(b\\right)\\right\\|$\u6709 $$\\frac{\\Phi_a\\left(c\\right)\\cdot\\Phi_a\\left(b\\right)}{\\Phi_a\\left(b\\right)\\cdot\\Phi_a\\left(b\\right)}=\\pm\\frac{\\left\\|\\Phi_a\\left(c\\right)\\right\\|}{\\left\\|\\Phi_a\\left(b\\right)\\right\\|}\\equiv\\alpha^\\prime$$ \u5176\u4e2d\u60c5\u51b51\uff09\u53d6\u6b63\u53f7\uff0c\u60c5\u51b52\uff09\u53d6\u8d1f\u53f7\u3002\u53e6\u5916\uff0c\u663e\u7136\u60c5\u51b53\uff09\u548c4\uff09$\\alpha^\\prime$\u7b49\u4e8e0\u6216\u65e0\u7a77\u3002\u6211\u4eec\u53ef\u4ee5\u91cd\u65b0\u6574\u7406\u56db\u79cd\u60c5\u51b5\uff0c\u5199\u6210\\begin{align*}&amp;\\Phi_a\\left(c\\right)\\cdot\\Phi_a\\left(b\\right)=\\alpha\\left(\\Phi_a\\left(b\\right)\\cdot\\Phi_a\\left(b\\right)\\right),\\alpha\\in\\mathbb{R}\\\\\\Leftarrow&amp;\\Phi_a\\left(c\\right)=\\alpha\\Phi_a\\left(b\\right)\\end{align*} \u4e0a\u5f0f\u4e2d\uff0c\u524d\u8005\u4ec5\u662f\u540e\u8005\u7684\u5fc5\u8981\u975e\u5145\u4efd\u6761\u4ef6\u3002\u5982\u679c\u8fd9\u4e00\u6b65\u662f\u5145\u8981\u6761\u4ef6\uff0c\u4ece\u800c\u5f97\u5230\u6700\u540e\u7684\u7ed3\u8bba\uff0c\u90a3\u4e48\u6211\u4eec\u65e0\u9700\u89c4\u5b9a$\\Phi_a$\u662f\u53cc\u5c04\uff0c\u5c31\u80fd\u628a$\\mathcal{E}$\u4e2d\u8fc7\u4efb\u610f\u4e24\u70b9$a,b$\u7684\u76f4\u7ebf\u4e0a\u7684\u70b9\u7684\u96c6\u5408\uff0c\u6620\u5c04\u4e3a$\\mathcal{V}$\u7684\u5b50\u96c6$\\mathcal{L}_{ab}=\\left\\{\\mathbf{c}|\\Phi_a\\left(c\\right)=\\alpha\\Phi_a\\left(b\\right),\\alpha\\in\\mathbb{R}\\}$\u3002\u4e8b\u5b9e\u4e0a\uff0c\u6613\u8bc1\u8fd9\u4e00\u5b50\u96c6\u662f$\\mathcal{V}$\u7684\u5b50\u7a7a\u95f4\u4e14\u7ef4\u6570\u662f1\u3002\u82e5\u6709\u4e86\u6b64\u57fa\u7840\uff0c\u5c31\u53ef\u8fdb\u4e00\u6b65\u63a8\u6f14\u51fa$\\mathcal{E}$\u7684\u7ef4\u6570\u5c31\u662f$\\mathcal{V}$\u7684\u7ef4\u6570\uff0c\u4ece\u5934\u5230\u5c3e\u65e0\u9700\u8bf4\u660e$\\Phi_a$\u662f\u53cc\u5c04\u3002\u53e6\u5916\uff0c\u8fd9\u4e5f\u540c\u65f6\u8bc1\u660e\u4e86\u53e6\u4e00\u4e2a\u91cd\u8981\u7684\u7ed3\u679c\uff0c\u5c31\u662f\u628a$\\mathcal{E}$\u4e2d\u7684\u76f4\u7ebf\u4e0e\u5b9e\u6570\u8f74\u5efa\u7acb\u4e86\u5bf9\u5e94\uff0c\u4e3a\u5efa\u7acb\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u5960\u5b9a\u4e86\u57fa\u7840\u3002\u53ef\u60dc\uff0c\u5728\u4e0a\u9762\u6211\u4eec\u53ea\u80fd\u5f97\u5230\u5fc5\u8981\u975e\u5145\u4efd\u6761\u4ef6\u7684\u5173\u7cfb\uff0c\u6240\u4ee5\u6211\u4eec\u5c31\u65e0\u6cd5\u62e5\u6709\u540e\u9762\u7684\u8fd9\u4e9b\u4f18\u7f8e\u53d9\u8ff0\u3002 \u5982\u679c\u6211\u4eec\u53cd\u8fc7\u6765\uff0c\u4ece$\\mathcal{V}$\u51fa\u53d1\u53bb\u627e$\\mathcal{E}$\u4e2d\u7684\u76f4\u7ebf\uff0c\u5c31\u514d\u4e0d\u4e86\u201c\u7ed9\u5b9a\u4efb\u4e00$\\mathbb{u}\\in\\mathcal{V}$\uff0c\u627e\u51fa\u5176\u5728$\\mathcal{E}$\u4e2d\u5bf9\u5e94\u7684\u6709\u5411\u7ebf\u6bb5\u201d\u7684\u4efb\u52a1\uff0c\u8fd9\u8981\u6c42\u6620\u5c04$\\Phi_a$\u662f\u53ef\u9006\u7684\uff0c\u5373\u4e3a\u53cc\u5c04\u3002\u867d\u7136\u6211\u770b\u4e0d\u5230$\\Phi_a$\u6709\u4ec0\u4e48\u7406\u7531\u4e0d\u80fd\u662f\u4e00\u4e2a\u53cc\u5c04\uff0c\u4f46\u6bd5\u7adf\u8fd9\u662f\u4e00\u9879\u91cd\u5927\u7684\u201c\u4ece$\\mathcal{V}$\u5230$\\mathcal{E}$\u7684\u53cd\u5411\u89c4\u5b9a\u201d\u52a8\u4f5c\uff0c\u8fd9\u6bd4\u524d\u9762\u7684\u201c\u957f\u5ea6\u201d\u548c\u201c\u89d2\u5ea6\u201d\u7684\u53cd\u5411\u89c4\u5b9a\u5f71\u54cd\u5927\u591a\u4e86\uff0c\u662f\u5168\u5c40\u6027\u5730\u89c4\u5b9a\u4e86$\\mathcal{E}$\u8fd9\u6574\u4e2a\u7a7a\u95f4\u3002\u5982\u6b64\u4e00\u6765\uff0c$\\mathcal{E}$\u4e0d\u4ec5\u5e26\u7740\u5176\u539f\u6709\u7684\u300a\u51e0\u4f55\u539f\u672c\u300b\u6240\u6709\u63a8\u8bba\uff0c\u53c8\u7a81\u7136\u4e8e\u4e00\u4e2a\u5411\u91cf\u7a7a\u95f4\u7b49\u8ddd\u540c\u6784\u3002\u51ed\u4ec0\u4e48\u5411\u91cf\u7a7a\u95f4\u2014\u2014\u4e00\u4e2a\u610f\u56fe\u5b8c\u5168\u4e0e\u51e0\u4f55\u65e0\u5173\u7684\u6570\u5b66\u6982\u5ff5\u2014\u2014\u6070\u597d\u91cd\u73b0\u4e86\u300a\u51e0\u4f55\u539f\u672c\u300b\u7684\u6b27\u51e0\u91cc\u5fb7\u7a7a\u95f4\u7684\u4e00\u5207\u6027\u8d28\u5462\uff1f\u6211\u627e\u4e0d\u5230\u5728\u8fd9\u4e2a\u5c42\u9762\u4e0a\u6709\u4ec0\u4e48\u4e07\u4f5b\u671d\u5b97\u5f0f\u7684\u89e3\u91ca\u3002\u5982\u679c\u6ca1\u6709\u8fd9\u79cd\u89e3\u91ca\uff0c\u90a3\u4e48\u5c31\u53ea\u80fd\u8ba4\u4e3a\uff0c\u8fd9\u4e00\u53cc\u5c04\u53ea\u662f\u91cd\u65b0\u5b9a\u4e49\u4e86\u4e00\u4e2a\u7a7a\u95f4$\\mathcal{E}$\uff0c\u5176\u51e0\u4f55\u6027\u8d28\u53ea\u7531\u5b9a\u4e49\u5b83\u7684\u90a3\u4e2a\u5185\u79ef\u7a7a\u95f4$\\mathcal{V}$\u6765\u89c4\u5b9a\u2014\u2014\u8fd9\u4e5f\u662f\u73b0\u4ee3\u51e0\u4f55\u5bf9\u201c\u6b27\u51e0\u91cc\u5fb7\u7a7a\u95f4\u201d\u7684\u5b9a\u4e49\uff0c\u5373\u5b8c\u5168\u629b\u5f03\u4e86\u300a\u51e0\u4f55\u539f\u672c\u300b\u3002\u65e2\u7136\u5982\u6b64\uff0c\u8fd9\u4e00\u73b0\u4ee3\u5b9a\u4e49\u7684\u201c\u6b27\u51e0\u91cc\u5fb7\u7a7a\u95f4\u201d\u4e2d\uff0c\u662f\u5426\u8981\u7528\u5411\u91cf\u4ee3\u6570\u91cd\u65b0\u8bc1\u660e\u4e00\u5207\u6570\u5b66\u53f2\u4e0a\u4ece\u300a\u51e0\u4f55\u539f\u672c\u300b\u8bc1\u660e\u8fc7\u7684\u51e0\u4f55\u5b9a\u7406\uff1f\u8fd9\u4e2a\u4efb\u52a1\u662f\u4e00\u4e2a\u5e9e\u5927\u7684\u4efb\u52a1\uff0c\u8fd8\u662f\u53ea\u9700\u91cd\u73b0\u51fa\u6b27\u51e0\u7406\u5fb7\u51e0\u4f55\u7684\u6709\u9650\u51e0\u6761\u516c\u8bbe\u5373\u53ef\uff1f\u8fd9\u4e2a\u4efb\u52a1\u662f\u5426\u5df2\u7ecf\u88ab\u5b8c\u6210\u4e86\uff1f\u5c3d\u7ba1\u5982\u6b64\uff0c\u6211\u4eec\u96be\u9053\u771f\u7684\u4e0d\u80fdrelax\u6389\u201c\u53cc\u5c04\u201d\u8fd9\u4e00\u89c4\u5b9a\u6027\u5417\uff1f<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"closed","ping_status":"open","sticky":false,"template":"","format":"aside","meta":{"inline_featured_image":false,"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":true,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[212],"tags":[244],"class_list":["post-8007","post","type-post","status-publish","format-aside","hentry","category-tagged","tag-mathematics","post_format-post-format-aside"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_sharing_enabled":true,"jetpack_shortlink":"https:\/\/wp.me\/p1aPvF-259","jetpack-related-posts":[{"id":9132,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/9132","url_meta":{"origin":8007,"position":0},"title":"\u4e0d\u57fa\u4e8e\u7cfb\u7efc\u6982\u5ff5\u7684\u5218\u7ef4\u5c14\u65b9\u7a0b\u63a8\u7406","author":"Andrew","date":"2025\u5e742\u670818\u65e5","format":false,"excerpt":"[latexpage] \u5218\u7ef4\u5c14\u65b9\u7a0b\u662f\u975e\u5e73\u8861\u6001\u7edf\u8ba1\u529b\u5b66\u4e2d\uff0c\u5173\u4e8e\u4efb\u610f\u7cfb\u7edf\u5fae\u72b6\u6001\u6982\u7387\u5bc6\u5ea6\u51fd\u6570$f\\left(\\bm\\Gamma,t\\right)$\u7684\u8fde\u7eed\u6027\u65b9\u7a0b\uff1a \\[\\frac{\\partial}{\\partial t}f+\\nabla_{\\bm\\Gamma}\\left(f\\dot{\\bm\\Gamma}\\right)=0\\] \u8fd9\u5b9e\u9645\u4e0a\u5728\u8bf4\uff0c\u4e0d\u4ec5\u4f5c\u4e3a\u6982\u7387\u5bc6\u5ea6\uff0c\u6709\u5f52\u4e00\u5316\u6761\u4ef6\u6240\u89c4\u5b9a\u7684\u5168\u57df\u6982\u7387\u5b88\u6052\\[\\int_{\\Lambda_t}f\\left(\\bm\\Gamma,t\\right)\\mathrm{d}\\bm\\Gamma\\equiv1,\\quad\\forall t\\]\u2014\u2014\u5176\u4e2d$\\Lambda_t\\in\\mathcal{B}$\u662f$t$\u65f6\u523b\u7cfb\u7edf\u53ef\u53d6\u7684\u6240\u6709\u72b6\u6001\u7684\u96c6\u5408\uff0c$\\mathcal{B}$\u8868\u793a$\\mathbb{R}^{6N}$\u4e0a\u7684Borel \u03c3-\u4ee3\u6570\u2014\u2014\u800c\u4e14\u5bf9\u4efb\u4e00\u5c40\u57df$\\omega_t\\subset\\Lambda_t,\\omega_t\\in\\mathcal{B}$\uff0c\u6982\u7387\u90fd\u5b88\u6052\u3002\u8fd9\u4ef6\u4e8b\uff0c\u5728\u4f20\u7edf\u6559\u79d1\u4e66\u5f53\u4e2d\u662f\u7528\u7cfb\u7efc\u7684\u6982\u5ff5\uff0c\u7136\u540e\u8bf4\u201c\u7cfb\u7efc\u7684\u7cfb\u7edf\u6570\u91cf\u4e0d\u53d8\u201d\u6765\u8ba8\u8bba\u7684\u3002\u4ee5\u4e0b\u6211\u4eec\u4ece\u6982\u7387\u8bba\u7684\u89d2\u5ea6\u5efa\u7acb\u7edf\u8ba1\u529b\u5b66\u7684\u57fa\u7840\u3002 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\u5728\u7ecf\u5178\u529b\u5b66\u5047\u5b9a\u4e0b\uff0c\u7cfb\u7edf\u7684\u72b6\u6001\u603b\u662f\u53ef\u4ee5\u7531\u6240\u6709\u7c92\u5b50\u7684\u4f4d\u7f6e\u548c\u52a8\u91cf\u6240\u786e\u5b9a\uff0c\u4e5f\u5c31\u662f\u8bf4\uff0c\u7cfb\u7edf\u7684\u72b6\u6001\u603b\u662f\u5bf9\u5e94\u7740$6N$\u7ef4\u5b9e\u6570\u7a7a\u95f4$\\mathbb{R}^{6N}$\u4e2d\u7684\u4e00\u4e2a\u70b9$\\bm\\Gamma\\equiv\\left(\\mathbf{r}^N,\\mathbf{p}^N\\right)$\uff0c\u5176\u4e2d\u6211\u4eec\u7b80\u8bb0$\\mathbf{r}^N\\equiv\\left(\\mathbf{r}_1,\\cdots,\\mathbf{r}_N\\right)$\u548c$\\mathbf{p}^N\\equiv\\left(\\mathbf{p}_1,\\cdots,\\mathbf{p}_N\\right)$\u3002 \u7531\u4e8e\u6211\u4eec\u65e0\u6cd5\u786e\u5b9a\u7cfb\u7edf\u5728\u4efb\u4f55\u65f6\u523b\u7684\u5fae\u89c2\u72b6\u6001\uff0c\u56e0\u6b64\u6211\u4eec\u6bcf\u4e00\u65f6\u523b\uff0c\u90fd\u4e3a\u7cfb\u7edf\u53ef\u53d6\u7684\u6bcf\u4e00\u4e2a\u53ef\u80fd\u72b6\u6001\u8d4b\u4ee5\u6982\u7387\u3002\u5177\u4f53\u5730\uff0c\u8bbe$\\left(\\mathbb{R}^{6N},\\mathcal{B}\\right)$\u662f\u4e00\u4e2aBorel\u53ef\u6d4b\u7a7a\u95f4\uff0c$\\Lambda_t\\in\\mathcal{B}$\u8868\u793a\u7cfb\u7edf\u5728$t$\u65f6\u523b\u6240\u6709\u53ef\u53d6\u7684\u5fae\u89c2\u72b6\u6001\u7684\u96c6\u5408\uff0c$\\mu_t:\\mathcal{B}\\rightarrow\\left[0,1\\right]$\u662f\u5b9a\u4e49\u5728\u8fd9\u4e2a\u53ef\u6d4b\u7a7a\u95f4\u4e0a\u7684\u6982\u7387\u6d4b\u5ea6\uff0c$\\mu_t\\left[X\\right],X\\in\\mathcal{B}$\u8868\u793a$t$\u65f6\u523b\u7cfb\u7edf\u7684\u5fae\u89c2\u72b6\u6001\u5728$X$\u5185\u7684\u6982\u7387\uff0c\u5b9e\u9645\u4e0a\\[\\mu_t\\left[X\\right]\\equiv\\int_Xf\\left(\\bm\\Gamma,t\\right)\\mathrm{d}\\bm\\Gamma\\]\u5176\u4e2d$f\\left(\\bm\\Gamma,t\\right)\\mathrm{d}\\bm\\Gamma$\u662f\u7cfb\u7edf\u5728$t$\u65f6\u523b\u5fae\u89c2\u72b6\u6001\u5904\u4e8e$\\bm\\Gamma\\sim\\bm\\Gamma+\\mathrm{d}\\Gamma$\u7684\u6982\u7387\u3002\u5141\u8bb8\u6982\u7387\u5bc6\u5ea6$f\\left(\\bm\\Gamma,t\\right)$\u5f97\u5230\u5b9a\u4e49\u6240\u9700\u8981\u7684$\\mu_t$\u5173\u4e8e$\\mathbb{R}^{6N}$\u7684\u52d2\u8d1d\u683c\u6d4b\u5ea6\u7684\u7edd\u5bf9\u8fde\u7eed\u6027\uff0c\u5b9e\u9645\u4e0a\u662f\u7531\u7ecf\u5178\u529b\u5b66\u672c\u8eab\u4fdd\u8bc1\u7684\uff0c\u89c1\u5173\u4e8e\u79bb\u6563\u6d4b\u5ea6\u6ca1\u6709\u5bc6\u5ea6\u7684\u95ee\u9898\u3002 \u5728\u7ecf\u5178\u529b\u5b66\u4e2d\uff0c\u7ed9\u5b9a\u67d0\u7cfb\u7edf\uff0c\u5b83\u7684\u8fd0\u52a8\u65b9\u7a0b\u603b\u662f\u628a\u4e00\u4e2a\u65f6\u523b$t$\u7684\u786e\u5b9a\u72b6\u6001$\\bm\\Gamma_t$\u6620\u5c04\u4e3a\u53e6\u4e00\u65f6\u523b$t+\\Delta t$\u7684\u552f\u4e00\u4e00\u4e2a\u786e\u5b9a\u72b6\u6001$\\bm\\Gamma_{t+\\Delta t}$\uff0c\u5373\u6211\u4eec\u53ef\u4ee5\u63d0\u53ca\u76f8\u5e94\u7684\u53cc\u5c04$\\chi_t:\\mathbb{R}^{6N}\\supset\\Lambda_t\\rightarrow\\mathbb{R}^{6N}$\u3002 \u82e5$\\Lambda_t\\in\\mathcal{B},\\omega_t\\in\\mathcal{B},\\omega_t\\subset\\Lambda_t,\\omega_{t+\\Delta t}=\\chi_t\\left(\\omega_t\\right)$\uff0c\u5219\u9996\u5148\u770b\u5230\u56e0\u4e3a$\\chi_t$\u5fc5\u662f\u8fde\u7eed\u7684\uff08\u7ecf\u5178\u529b\u5b66\u8fd0\u52a8\uff09\uff0c\u6545$\\chi_t$\u603b\u628a\u5f00\u96c6\u6620\u5c04\u4e3a\u5f00\u96c6\uff0c\u5373$\\omega_{t+\\Delta t}\\in\\mathcal{B}$\uff0c\u6545$\\chi_t$\u662f\u53ef\u6d4b\u6620\u5c04\u3002\u5176\u6b21\uff0c\u6211\u4eec\u770b\u5230\uff1a\\[\\mu_{t+\\Delta t}\\left(\\omega_{t+\\Delta t}\\right)=\\chi_t_*\\mu_t\\left(\\omega_t\\right)=\\mu_t\\left(\\chi_t^{-1}\\left(\\omega_{t+\\Delta t}\\right)\\right)=\\mu_t\\left(\\omega_t\\right)\\]\u5176\u4e2d\u8bb0\u53f7$f_*\\mu$\u8868\u793a\u6d4b\u5ea6$\\mu_t$\u7684\u7531\u53ef\u6d4b\u6620\u5c04$f$\u5f15\u51fa\u7684\u524d\u63a8\u6d4b\u5ea6\uff08pushforward measrue\uff09\uff0c\u4e0a\u9762\u7b2c\u4e8c\u4e2a\u7b49\u5f0f\u7528\u5230\u4e86\u524d\u63a8\u6d4b\u5ea6\u7684\u5b9a\u4e49\uff0c\u7b2c\u4e09\u4e2a\u7b49\u5f0f\u7528\u5230\u4e86$\\chi_t$\u7684\u53cc\u5c04\u6027\u3002\u4ee5\u4e0a\u63a8\u7406\u5bf9\u4efb\u610f$t,\\Delta t\\in\\mathbb{R}$\u4ee5\u53ca\u4efb\u610f$\\omega_t$\u5747\u6210\u7acb\u3002\u56e0\u6b64\u6211\u4eec\u8bc1\u660e\u4e86\uff0c$\\mathbb{R}^{6N}$\u7a7a\u95f4\u4efb\u4e00\uff08\u53ef\u6d4b\uff09\u5c40\u57df\u4e0a\u7684\u6982\u7387\u662f\u6cbf\u533a\u57df\u968f\u6309\u7167\u5b9e\u9645\u7cfb\u7edf\u8fd0\u52a8\u7684\u8fd0\u52a8\u7684\u5b88\u6052\u91cf\u3002\u56e0\u6b64\u6709\\[\\frac{\\mathrm{d}}{\\mathrm{d}t}\\mu_t\\left[\\omega_t\\right]\\equiv 0\\] \u540e\u7eed\u7531\u96f7\u8bfa\u4f20\u8f93\u5b9a\u7406\u5373\u53ef\u5f97\u5230\u5173\u4e8e\u6982\u7387\u5bc6\u5ea6\u51fd\u6570$f\\left(\\bm\\Gamma,t\\right)$\u7684\u5218\u7ef4\u5c14\u65b9\u7a0b\u3002","rel":"","context":"In &quot;\u6211\u7684\u7814\u7a76\u7b14\u8bb0&quot;","block_context":{"text":"\u6211\u7684\u7814\u7a76\u7b14\u8bb0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/rheo"},"img":{"alt_text":"","src":"","width":0,"height":0},"classes":[]},{"id":43,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/43","url_meta":{"origin":8007,"position":1},"title":"\u6211\u7684\u535a\u58eb\u8bfe\u9898\u7684\u79d1\u666e\u4ecb\u7ecd","author":"Andrew","date":"2008\u5e7410\u670825\u65e5","format":false,"excerpt":"\u6211\u7684\u535a\u58eb\u8bfe\u9898\u7684\u79d1\u666e\u4ecb\u7ecd \u6211\u5728\u8fd9\u91cc\u4ecb\u7ecd\u4e00\u4e0b\u6211\u7684\u535a\u58eb\u8bfe\u9898\u7684\u80cc\u666f\u3002 \u81ea\u76f8\u4f3c\u2014\u2014\u5927\u81ea\u7136\u7684\u827a\u672f Why is geometry often described as 'cold' and 'dry'? One reason lies in its inability to describe the shape of a cloud, a mountain, a coastline or a tree. Clouds are not spheres, mountains are not cones, coastlines are not circles, and bark is not smooth, nor\u2026","rel":"","context":"In &quot;\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0&quot;","block_context":{"text":"\u4ee5tag\u5206\u7c7b\u7684\u6587\u7ae0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/tagged"},"img":{"alt_text":"","src":"http:\/\/docs.google.com\/File?id=ad9bz9mnxsph_695fz26xxdr_b","width":350,"height":200},"classes":[]},{"id":5377,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/5377","url_meta":{"origin":8007,"position":2},"title":"Fourier\u53d8\u6362\uff08\u4e09\uff09","author":"Andrew","date":"2013\u5e745\u67082\u65e5","format":false,"excerpt":"[mathjax] \u901a\u8fc7\u4e0a\u4e00\u7bc7\u6211\u4eec\u5df2\u7ecf\u5b66\u4f1a\u600e\u6837\u901a\u8fc7Fourier\u53d8\u6362\u91cd\u6784\u5e94\u53d8\u548c\u5e94\u529b\u7684\u6ce2\u5f62\u3002\u81f3\u5c11In\/1\u662f\u5df2\u7ecf\u53ef\u4ee5\u76f4\u63a5\u7528Fourier\u53d8\u6362\u7684\u7ed3\u679c\u8ba1\u7b97\u4e86\u3002\u672c\u6587\u5148\u4ecb\u7ed3\u5b9e\u9645\u5b9e\u9a8c\u6570\u636e\u7684\u5bfc\u5165\uff0c\u518d\u7ee7\u7eed\u4e0a\u4e00\u7bc7\u7684\u8bdd\u9898\u4ecb\u7ecd\u5982\u4f55\u8ba1\u7b97In\/1\u548c\u5176\u4f59\u7684LAOS\u53c2\u6570\u3002 LAOS\u5b9e\u9a8c\u662f\u600e\u4e48\u5b9e\u65bd\u7684 \u6211\u4eec\u662f\u5728TA ARES-RFS\u6d41\u53d8\u4eea\u4e0a\u505aLAOS\u5b9e\u9a8c\u3002\u516c\u53f8\u7684\u540d\u5b57\uff1aTA Instruments\u3002\u8fd9\u662f\u4e00\u53f0\u5e94\u53d8\u63a7\u5236\u578b\u6d41\u53d8\u4eea\uff08strain-controlled 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sweep\u662f\u56fa\u5b9a\u632f\u8361\u7684\u5e94\u53d8\u5e45\u5ea6\u03b30\u548c\u9891\u7387\u03c9\uff0c\u6d4b\u91cfG'\u548cG''\u968f\u65f6\u95f4\u53d8\u5316\u7684\u6d4b\u8bd5\uff0c\u6240\u4ee5\u6b63\u9002\u5408\u6211\u4eec\u3002\u5728Orchestrator\u8f6f\u4ef6\u4e2d\u8bbe\u5b9a\u597ddynamic time sweep\u7684\u5e94\u53d8\u548c\u9891\u7387\uff0c\u4eea\u5668\u5c31\u4f1a\u4ee5\u76f8\u5e94\u7684\u6b63\u5f26\u51fd\u6570\u53bb\u63a7\u5236\u9a6c\u8fbe\u4e86\u3002\u6b64\u65f6\uff0c\u6211\u4eec\u9700\u8981\u9a6c\u4e0a\u4ece\u4eea\u5668\u80cc\u540e\u7684\u6a21\u62df\u8f93\u51fa\u7aef\u53e3\u540c\u65f6\u8bfb\u51fa\u4eea\u5668\u7684\u6d4b\u91cf\u4fe1\u53f7\uff0c\u5e76\u7528\u4e00\u4e2a\u6570\u6a21\u8f6c\u6362\u5668\uff08analog-to-digital converter, ADC\uff09\u8f6c\u6362\u6210\u6570\u5b57\u4fe1\u53f7\uff0c\u8f93\u5165\u5230\u8ba1\u7b97\u673a\u3002\u6211\u4eec\u7528\u7684ADC\u662fNI USB-9215A\uff0c\u8981\u901a\u8fc7\u7535\u8111\u6765\u63a7\u5236\u5b83\u4f55\u65f6\u5f00\u59cb\u548c\u505c\u6b62\u91c7\u6837\uff0c\u4ee5\u53ca\u4ee5\u591a\u5927\u7684\u91c7\u6837\u9891\u7387\u8fdb\u884c\u91c7\u6837\u3002\u8bb8\u591a\u8f6f\u4ef6\u90fd\u53ef\u4ee5\u901a\u8fc7\u5b83\u7684\u9a71\u52a8\u7a0b\u5e8f\u63a5\u53e3\u6765\u5b9e\u65bd\u63a7\u5236\uff08MATLAB\u672c\u8eab\u5c31\u53ef\u4ee5\uff09\uff0c\u5728LAOS\u5b9e\u9a8c\u4e2d\u6211\u4eec\u7528\u4e00\u4e2aLabView\u5f00\u53d1\u7684\u5de5\u5177FT Meassurement\u6765\u91c7\u96c6\u6570\u636e\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u7528\u4e00\u53f0\u7535\u8111\uff0c\u7528\u4e24\u4e2a\u8f6f\u4ef6\u5206\u800c\u63a7\u5236\u4e0d\u540c\u7684\u8fc7\u7a0b\u3002Orchestrator\u63a7\u5236\u6d41\u53d8\u4eea\u7684\u8fd0\u4f5c\uff0c\u800cFT Meassurement\u63a7\u5236ADC\u3002\u8fd9\u4e00\u6d41\u7a0b\u56fe\u53ef\u53c2\u8003\u548c\u4e2d\u7684\u7528\u56fe\u3002\u5177\u4f53\u4fe1\u53f7\u7ebf\u63a5\u6cd5\u548c\u5904\u7406\u65b9\u5f0f\u89c1\u4e0b\u9762\u7684PPT\uff08\u53ef\u4e0b\u8f7d\uff09\u3002 ARES\u6a21\u62df\u8f93\u51fa View more presentations or Upload your own. \u4e0a\u7bc7\u6587\u7ae0\u4e5f\u8bf4\u8fc7\uff0c\u4ece\u6d41\u53d8\u4eea\u91c7\u96c6\u5230\u7684\u4e00\u4e2a\u662f\u9a6c\u8fbe\u4f4d\u79fb\u7684\u7535\u538b\u4fe1\u53f7\uff0c\u53e6\u4e00\u4e2a\u662f\u529b\u77e9\u7684\u7535\u538b\u4fe1\u53f7\uff0c\u5b83\u4eec\u90fd\u8981\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u624d\u662f\u5e94\u53d8\u548c\u5e94\u529b\u3002\u4eceARES\u6d41\u53d8\u4eea\u7684\u8bf4\u4e66\u4e2d\u53ef\u4ee5\u67e5\u5230\u80cc\u677f\u8f93\u51fa\u4fe1\u53f7\u7684\u5177\u4f53\u60c5\u51b5\uff1a TORQUE OUT\uff1a\u4ece0\u5230\u6ee1\u91cf\u7a0b\uff0c\u5bf9\u5e94\u7535\u538b\u662f0 ~ 5 V\u3002\u7531\u4e8e\u8f6c\u77e9\u53ef\u4ee5\u6709\u4e24\u4e2a\u65b9\u5411\uff0c\u6240\u4ee5\u603b\u7684\u7535\u538b\u8303\u56f4\u5c31\u662f-5 ~ +5 V STRAIN OUT\uff1a\u867d\u7136\u6807\u8bb0\u662fSTRAIN\uff0c\u4f46\u5176\u5b9e\u53ea\u662f\u9a6c\u8fbe\u8f6c\u52a8\u7684\u89d2\u4f4d\u79fb\u3002\u4ece0 ~ 0.5 rad\u5bf9\u5e94\u7535\u538b\u662f0~5V\uff0c\u8003\u8651\u4e24\u4e2a\u8f6c\u52a8\u65b9\u5411\uff0c\u603b\u7535\u538b\u8303\u56f4\u662f-5~+5 V TORQUE OUT\u7684\u201c\u6ee1\u91cf\u7a0b\u201d\u5230\u5e95\u662f\u591a\u5c11\u5462\uff1f\u7531\u4e8e\u6211\u4eec\u7684ARES\u6709\u4e24\u4e2a\u6863\u4f4d\u7684\u529b\u77e9\u4f20\u611f\u5668\u3002Xducer 1\u7684\u4f20\u611f\u5668\u53eb\u505a\u201c20g FRT\u201d\u300220g\u662f\u6307\u8be5\u6863\u4f4d\u7684\u6700\u5927\u91cf\u7a0b\u662f20 g\u00b7cm\uff0c\u529b\u77e9\u7684\u4e00\u4e2a\u975eSI\u5355\u4f4d\uff0cSI\u5355\u4f4d\u662fN\u00b7m\u30021 g\u00b7cm =\u2026","rel":"","context":"In 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\u5927\u5e45\u632f\u8361\u526a\u5207\u7684\u53c2\u6570 \u5927\u5e45\u632f\u8361\u526a\u5207\u5b9e\u9a8c\u4ea7\u751f\u7684\u6ce2\u5f62\u662f\u975e\u6b63\u5f26\u4fe1\u53f7\uff0c\u975e\u6b63\u5f26\u662f\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7279\u70b9\u3002\u6240\u4ee5\u5bf9\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7a0b\u5ea6\u7684\u8868\u5f81\uff0c\u5c31\u662f\u5bf9\u6ce2\u5f62\u6b63\u5f26\u5f62\u72b6\u504f\u79bb\u7a0b\u5ea6\u7684\u8868\u5f81\u3002\u5173\u4e8e\u5927\u5e45\u632f\u8361\u526a\u5207\u548c\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u539f\u7406\u4e0e\u5b9e\u8df5\uff0c\u53ef\u89c1Hyhn\u7b49\u7684\u7efc\u8ff0\uff0c\u6216\u8005\u53c2\u9605\u6211\u7684\u535a\u58eb\u5b66\u4f4d\u8bba\u6587\u7b2c1.2.1\u8282\u3001\u7b2c1.3\u8282\u548c\u7b2c2\u7ae0\u3002\u672c\u6587\u53ea\u89e3\u51b3MATLAB\u7f16\u7a0b\u5b9e\u73b0\u95ee\u9898\u3002 \u5927\u5e45\u632f\u8361\u526a\u5207\uff08LAOS\uff09\u5b9e\u9a8c\u4e2d\uff0c\u6211\u4eec\u4ece\u6d41\u53d8\u4eea\u540c\u65f6\u83b7\u53d6\u4e24\u4e2a\u4fe1\u53f7\u3002\u4e00\u4e2a\u662f\u5e94\u53d8\u7684\u4fe1\u53f7\uff0c\u4e00\u4e2a\u662f\u5e94\u529b\u7684\u4fe1\u53f7\uff08\u4e8b\u5b9e\u4e0a\uff0c\u4e00\u4e2a\u662f\u9a6c\u8fbe\u4f4d\u79fb\u4fe1\u53f7\uff0c\u53e6\u4e00\u4e2a\u662f\u529b\u77e9\u4fe1\u53f7\uff0c\u5b83\u4eec\u90fd\u8981\u4e58\u4ee5\u4e00\u4e2a\u7cfb\u6570\uff0c\u624d\u662f\u5e94\u53d8\u548c\u5e94\u529b\u3002\u6b64\u6587\u5148\u7565\u53bb\u8fd9\u4e00\u4e8b\u5b9e\u4e86\uff09\u3002\u5728\u5e94\u53d8\u63a7\u5236\u7684\u5b9e\u9a8c\u4e2d\uff0c\u6211\u4eec\u65bd\u52a0\u4e00\u4e2a\u6b63\u5f26\u7684\u5e94\u53d8\uff0c\u6d4b\u91cf\u5e94\u529b\u7684\u54cd\u5e94\uff0c\u5373 $$\\begin{cases}\\gamma\\left(t\\right)=\\sin\\omega t\\\\ \\sigma\\left(t\\right)=\\sum_{n=0:\\text{odd}}^\\infty\\sigma_n\\sin\\left(\\omega t+\\delta_n\\right)\\end{cases}$$\u7531\u6b64\u5f0f\u53ef\u5b9a\u4e49LAOS\u5b9e\u9a8c\u7ed3\u679c\u7684\u8868\u5f81\u53c2\u6570\uff0c\u4ee5\u53ca\uff0c\u6765\u8868\u5f81\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7a0b\u5ea6\uff08\u5373\u504f\u79bb\u6b63\u5f26\u51fd\u6570\u7684\u7a0b\u5ea6\u3002\u6b64\u5916\uff0c\u6211\u4eec\u8fd8\u4f1a\u4ee5\u5bf9\u4f5c\u56fe\uff08\u53c2\u6570\u65b9\u7a0b\u4f5c\u56fe\uff09\uff0c\u5f97\u5230Lissajous\u66f2\u7ebf\u3002\u7136\u540e\u7528\u66f2\u7ebf\u7684\u5f62\u72b6\u7279\u5f81\u6765\u8868\u5f81\u975e\u7ebf\u6027\u7c98\u5f39\u6027\u7684\u7a0b\u5ea6\uff08\u5373\u66f2\u7ebf\u504f\u79bb\u692d\u5706\u7684\u7a0b\u5ea6\uff09\u3002\u63cf\u8ff0\u66f2\u7ebf\u5f62\u72b6\u7279\u5f81\uff0c\u5f80\u5f80\u5c31\u662f\u5404\u79cd\u5404\u6837\u7684\u5207\u7ebf\u659c\u7387\uff08\u53ef\u81ea\u884c\u4e0a\u7f51\u641c\u7d22\u590d\u4e60\u53c2\u6570\u65b9\u7a0b\u6c42\u5bfc\uff09\u3002\u5728LAOS\u4e2d\u5e38\u7528\uff1a$$G_\\text{M}=\\left.\\frac{d\\sigma}{d\\gamma}\\right|_{\\gamma=0}$$$$G_\\text{L}=\\frac{\\sigma{\\gamma_0}}{\\gamma_0}$$$$G_\\text{k}=\\left.\\frac{d\\sigma}{d\\gamma}\\right|_{\\sigma=0}$$ MATLAB\u7f16\u7a0b\u6b65\u9aa4 \u5728\u8be6\u7ec6\u4ecb\u7ecdMATLAB\u4ee3\u7801\u4e4b\u524d\uff0c\u5148\u603b\u7ed3\u4e00\u4e0b\u6211\u4eec\u8981\u5b8c\u6210\u4ec0\u4e48\u4efb\u52a1\uff0c\u8ddf\u4ece\u54ea\u4e9b\u6b65\u9aa4\u3002\u6211\u4eec\u7684\u4efb\u52a1\u5c31\u662f\u8981\u4ece\u5e94\u529b\u548c\u5e94\u53d8\u7684\u4fe1\u53f7\u8ba1\u7b97\u51fa\u4e0a\u8ff0\u7684\u8fd9\u4e48\u591a\u53c2\u6570\u3002\u5b9e\u9a8c\u65f6\uff0c\u6211\u4eec\u4f1a\u89c4\u5b9a\u5e94\u53d8\u7684\u5e45\u5ea6\u548c\u9891\u7387\uff0c\u4f46\u662f\uff0c\u7531\u4e8e\u4eea\u5668\u8bef\u5dee\uff0c\u5b9e\u9645\u65bd\u52a0\u5728\u6837\u54c1\u4e0a\u7684\u5e94\u53d8\u5e45\u5ea6\u548c\u9891\u7387\uff0c\u672a\u5fc5\u5b8c\u5168\u7b49\u4e8e\u7684\u547d\u4ee4\u503c\u3002\u5c3d\u7ba1\u8fd9\u4e00\u8bef\u5dee\u975e\u5e38\u5c0f\uff0c\u4f46\u4ece\u505a\u6cd5\u4e0a\u6211\u4eec\u8fd8\u662f\u8981\u4ee5\u5b9e\u9645\u5e94\u53d8\u4fe1\u53f7\u4e3a\u51c6\u3002\u6240\u4ee5\uff0c\u6211\u4eec\u9996\u5148\u8981\u5bf9\u5b9e\u9645\u5e94\u53d8\u4fe1\u53f7\u8fdb\u884c\u5224\u65ad\uff0c\u83b7\u5f97\u5b9e\u9645\u7684\u5e94\u53d8\u5e45\u5ea6\u548c\u9891\u7387\u3002\u5982\u4f55\u5224\u65ad\u4e00\u4e2a\u672a\u77e5\u6b63\u5f26\u4fe1\u53f7\u7684\u632f\u5e45\u3001\u9891\u7387\u548c\u76f8\u4f4d\u7684\u65b9\u6cd5\u5df2\u7ecf\u5728\u4e0a\u4e00\u7bc7\u6587\u7ae0\u4e2d\u4ecb\u7ecd\u4e86\u3002 \u7136\u540e\uff0c\u5c31\u8981\u5904\u7406\u5e94\u529b\u7684\u4fe1\u53f7\u3002\u4e3a\u4e86\u83b7\u5f97\u548c\u540c\u6837\u9700\u8981\u5bf9\u8fdb\u884cfft\u5904\u7406\uff0c\u65b9\u6cd5\u8ddf\u4e0a\u4e00\u7bc7\u6587\u7ae0\u4ecb\u7ecd\u7684\u7c7b\u4f3c\uff0c\u533a\u522b\u53ea\u5728\u4e8e\u6211\u4eec\u9664\u4e86\u8981\u83b7\u5f97\u57fa\u9891\u7684\u5e45\u503c\u548c\u76f8\u4f4d\u4e4b\u5916\uff0c\u8fd8\u8981\u83b7\u5f97\u9ad8\u6b21\u8c10\u6ce2\u7684\u5e45\u503c\u548c\u76f8\u4f4d\u3002\u7406\u8bba\u4e0a\u6765\u8bf4\uff0c\u4f46\u662f\u5b9e\u9645\u4e0a\u6211\u4eec\u6839\u636e\u9700\u8981\u53d6\u6709\u9650\u7684n\u4e2a\u9ad8\u6b21\u8c10\u6ce2\uff0c\u5c31\u53ef\u4ee5\u6bd4\u8f83\u51c6\u786e\u5730\u91cd\u6784\u51fa\u7684\u6ce2\u5f62\u4e86\u3002\u63d0\u9192\u4e00\u4e0b\uff0c\u5b9a\u4e49\u5f0f\u91cc\u5e94\u53d8\u662f\u4e00\u4e2a\u96f6\u76f8\u4f4d\u7684\u6807\u51c6\u6b63\u5f26\u51fd\u6570\uff0c\u4f46\u5b9e\u9645\u91cd\u6784\u7684\u5e94\u53d8\u6ce2\u5f62\u53ef\u80fd\u4f1a\u6709\u975e\u96f6\u7684\u76f8\u4f4d\uff0c\u8fd9\u65f6\uff0c\u5e94\u529b\u7684\u5404\u4e2a\u8c10\u6ce2\u7684\u76f8\u4f4d\u8981\u90fd\u51cf\u53bb\u5e94\u53d8\u7684\u76f8\u4f4d\uff0c\u624d\u662f\u6211\u4eec\u8981\u6c42\u7684\u3002 Lissajous\u66f2\u7ebf\u7684\u5404\u79cd\u5bfc\u6570\uff0c\u8981\u540c\u65f6\u5728\u548c\u90fd\u91cd\u6784\u4e86\u4e4b\u540e\uff0c\u518d\u8fdb\u884c\u6c42\u5bfc\u8ba1\u7b97\u5f97\u5230\u3002\u603b\u800c\u8a00\u4e4b\uff0c\u8981\u5b8c\u6210\u4ee5\u4e0b\u4e09\u4e2a\u6b65\u9aa4\uff1a \u91cd\u6784\u5e94\u53d8\uff0c\u83b7\u5f97\u3001\u548c\u76f8\u4f4d \u91cd\u6784\uff0c\u83b7\u5f97\u548c\uff0c\u8ba1\u7b97 \u7528\u91cd\u6784\u7684\u548c\u4f5cLissajous\u66f2\u7ebf\uff0c\u4ee5\u53ca\u8ba1\u7b97\u3001\u548c \u91cd\u6784\u548c MATLAB\u5e2e\u52a9\u6587\u4ef6\u91ccfft\u547d\u4ee4\u7684\u6761\u76ee\u6709\u4e00\u4e2a\u8303\u4f8b\uff0c\u5e94\u8be5\u80fd\u89e3\u51b3\u5927\u90e8\u5206\u95ee\u9898\u4e86\u3002\u5176\u4e2d\u5728coherent sampling\uff08\u89c1\u4e0a\u7bc7\u6587\u7ae0\uff09\u5904\uff0c\u8303\u4f8b\u7528next power of 2\uff0c\u5176\u5b9e\u6ca1\u6709\u5fc5\u8981\u3002\u4e0b\u9762\u7684\u4ee3\u7801\uff0c\u6211\u5c31\u4e0d\u591a\u4f5c\u89e3\u91ca\u4e86\u3002 clear all clc % the strain signal fs=1000; % sampling frequency t=0:1\/fs:30.2; % the time series of the strain signal t=t(:); % convert to colume vector w=5; %\u2026","rel":"","context":"In &quot;\u5de5\u4f5c\u7b14\u8bb0&quot;","block_context":{"text":"\u5de5\u4f5c\u7b14\u8bb0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/personal\/%e5%b7%a5%e4%bd%9c%e7%ac%94%e8%ae%b0"},"img":{"alt_text":"Figure 1 Parts of the strain and the stress signals","src":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/fig1.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/fig1.png?resize=350%2C200 1x, https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/fig1.png?resize=525%2C300 1.5x"},"classes":[]},{"id":5244,"url":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/5244","url_meta":{"origin":8007,"position":5},"title":"Fourier\u53d8\u6362\uff08\u4e00\uff09","author":"Andrew","date":"2013\u5e744\u67088\u65e5","format":false,"excerpt":"[mathjax] \u4e3a\u4e86\u5e26\u5e08\u5f1f\u505a\u5b9e\u9a8c\uff0c\u7b80\u8981\u603b\u7ed3\u4e00\u4e0b\u5728MATLAB\u91cc\u8fdb\u884cFourier\u53d8\u6362\u7684\u4e00\u4e9b\u4ee3\u7801\uff0c\u4ee5\u4f9b\u5927\u5bb6\u53c2\u8003\u3002 Discrete Fourier transform (DFT): The sequence of N complex numbers x0, ..., xN-1 is transformed into an N-periodic sequence of complex numbers according to the DFT formula . \u4e00\u822c\u628a\u53d8\u6362\u524d\u7684\u4fe1\u53f7xn\u5f53\u6210\u65f6\u57df\u4fe1\u53f7\uff0c\u53d8\u6362\u540e\u7684Xk\u5f53\u6210\u9891\u57df\u4fe1\u53f7\u3002\u53ef\u89c1\uff0c\u65f6\u57df\u662fN\u4e2a\u5355\u5143\u7684\u4fe1\u53f7\uff0c\u53d8\u6362\u540e\u9891\u57df\u8fd8\u662fN\u4e2a\u5355\u5143\u7684\u4fe1\u53f7\u3002\u4ec0\u4e48\u5468\u671f\u554a\u9891\u7387\u554a\u8fd9\u4e9b\u4fe1\u606f\u662f\u6682\u65f6\u6ca1\u6709\u7684\u3002\u4f8b\u5982xn\u4e3a\u4ee5\u4e0b\u8fd9\u4e2a\u5e8f\u5217\uff08Fig. 1\uff09\uff0c\u6a2a\u5750\u6807\u5c31\u662f\u5e8f\u53f7n\u3002\u8fd9\u6bb5\u5e8f\u5217\uff0c\u53ef\u4ee5\u662f\u4e00\u6bb510\u79d2\u949f\u7684\u7684\u4fe1\u53f7\uff0c\u4e5f\u53ef\u4ee5\u662f\u4e00\u6bb5100\u79d2\u949f\u7684\u4fe1\u53f7\u3002\u5f53\u7136\uff0c\u5b9e\u9645\u60c5\u51b5\u4e0b\u4e00\u6bb5\u4fe1\u53f7\u662f\u591a\u5c11\u79d2\u662f\u4e8b\u5148\u77e5\u9053\u7684\uff0c\u51fa\u6765\u7684\u4fe1\u53f7\u4e00\u5171\u6709\u591a\u5c11\u4e2a\u5355\u5143\u503c\uff08N\uff09\u5219\u53d6\u51b3\u4e8e\u6211\u4eec\u9694\u591a\u957f\u65f6\u95f4\u53d6\u4e00\u6b21\u6837\u3002\u5982\u679c\u4e00\u6bb510\u79d2\u949f\u7684\u4fe1\u53f7\uff0c\u6211\u4eec\u6bcf\u79d2\u53d6\u4e00\u6b21\u6837\uff0c\u8fd9\u4e2a\u5e8f\u5217\u5c31\u670910\u4e2a\u5355\u5143\uff08N = 10\uff09\u3002\u53cd\u4e4b\uff0cFig. 1\u7684\u8fd9\u4e2a\u5e8f\u5217N = 93\uff0c\u5047\u5982\u8fd9\u4e2a\u4fe1\u53f7\u662f\u6bcf\u96940.1\u79d2\u53d6\u4e00\u6b21\u6837\u5f97\u5230\u7684\uff0c\u8fd9\u4e2a\u4fe1\u53f7\u7684\u603b\u65f6\u957f\u5c31\u662f9.2\u79d2\uff1b\u800c\u5047\u5982\u8fd9\u4e2a\u4fe1\u53f7\u662f\u6bcf\u969410\u79d2\u53d6\u4e00\u6b21\u6837\u5f97\u5230\u7684\uff0c\u8fd9\u4e2a\u4fe1\u53f7\u603b\u65f6\u957f\u5c31\u662f920\u79d2\u4e86\u3002\u6bcf\u9694\u591a\u957f\u65f6\u95f4\u53d6\u4e00\u6b21\u6837\uff0c\u53ef\u4ee5\u7528\u53d6\u6837\u9891\u7387fs\u6765\u8868\u793a\u3002fs = 10 Hz\u5c31\u662f\u6bcf\u96940.1 s\u53d6\u4e00\u6b21\u6837\u3002\u6240\u4ee5\uff0c\u4e3a\u4e86\u7ed9\u4e00\u5bf9\u79bb\u6563Fourier\u53d8\u6362xn\u548cXk\u8d4b\u4ee5\u5468\u671f\u6216\u9891\u7387\u7684\u4fe1\u606f\uff0c\u5fc5\u987b\u7ed9\u65f6\u57df\u4fe1\u53f7\u89c4\u5b9a\u4e00\u4e2afs\u3002\u4e00\u65e6fs\u89c4\u5b9a\u4e86\uff0c\u5468\u671f\u548c\u9891\u7387\u4e5f\u5c31\u89c4\u5b9a\u4e86\u3002\u8fd8\u62ffFig. 1\u7684\u5e8f\u5217\u4e3a\u4f8b\uff0c\u8bbe\u8be5\u5e8f\u5217\u7684\u53d6\u6837\u9891\u7387fs = 10 Hz\uff0c\u5219\u8be5\u5e8f\u5217\u6bcf\u4e2a\u5355\u5143\u9694\u4e860.1 s\u3002\u6211\u4eec\u53ef\u4ee5\u6570\u4e00\u4e0b\u8fd9\u4e2a\u5e8f\u5217\u4e00\u4e2a\u5468\u671f\u6709\u591a\u5c11\u4e2a\u5355\u5143\u2014\u201420\u4e2a\uff0c\u90a3\u4e48\u8be5\u5e8f\u5217\u7684\u5468\u671fT = 2 s\uff0c\u9891\u7387f =\u2026","rel":"","context":"In &quot;\u5de5\u4f5c\u7b14\u8bb0&quot;","block_context":{"text":"\u5de5\u4f5c\u7b14\u8bb0","link":"https:\/\/www.andrewsun.net\/panta_rhei\/archives\/category\/personal\/%e5%b7%a5%e4%bd%9c%e7%ac%94%e8%ae%b0"},"img":{"alt_text":"Figure 1","src":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/f1.png?resize=350%2C200","width":350,"height":200,"srcset":"https:\/\/i0.wp.com\/www.andrewsun.net\/panta_rhei\/wp-content\/uploads\/2013\/04\/f1.png?resize=350%2C200 1x, 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