{"id":39449,"date":"2011-09-13T17:25:13","date_gmt":"2011-09-13T09:25:13","guid":{"rendered":"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/?p=39449"},"modified":"2021-10-06T16:22:48","modified_gmt":"2021-10-06T08:22:48","slug":"%e8%aa%8d%e8%ad%98%e7%ad%89%e8%a7%92%e8%9e%ba%e7%b7%9aon-the-equiangular-spiral","status":"publish","type":"post","link":"http:\/\/localhost\/%e8%aa%8d%e8%ad%98%e7%ad%89%e8%a7%92%e8%9e%ba%e7%b7%9aon-the-equiangular-spiral\/","title":{"rendered":"\u8a8d\u8b58\u7b49\u89d2\u87ba\u7dda(On the Equiangular Spiral)"},"content":{"rendered":"<div class=\"pf-content\"><p><strong><span style=\"color: #ff6600;\">\u8a8d\u8b58\u7b49\u89d2\u87ba\u7dda(On the Equiangular Spiral)<\/span><br \/>\n<span style=\"color: #008000;\">\u570b\u7acb\u81fa\u7063\u5e2b\u7bc4\u5927\u5b78\u6578\u5b78\u7cfb\u8d99\u6587\u654f\u6559\u6388\/\u570b\u7acb\u81fa\u7063\u5e2b\u7bc4\u5927\u5b78\u6578\u5b78\u7cfb\u8d99\u6587\u654f\u6559\u6388\u8cac\u4efb\u7de8\u8f2f<\/span><\/strong><\/p>\n<p><strong>\u6458\u8981:\u672c\u6587\u4ecb\u7d39\u7b49\u89d2\u87ba\u7dda\u7684\u6b77\u53f2\u8207\u4e00\u4e9b\u6027\u8cea\u3002<\/strong><\/p>\n<p><span style=\"color: #000080;\"><strong>\u4f55\u8b02\u7b49\u89d2\u87ba\u7dda<\/strong><\/span><\/p>\n<p>\u5728\u4e00\u7247\u7a7a\u66e0\u7684\u8349\u5730\u4e0a\uff0c\u7532\u3001\u4e59\u3001\u4e19\u3001\u4e01\u56db\u96bb\u72d7\u5206\u5225\u7ad9\u7acb\u5728\u4e00\u500b\u6b63\u65b9\u5f62\u7684\u56db\u500b\u9802\u9ede \\(A\\)\u3001\\(B\\)\u3001\\(C\\)\u3001\\(D\\) \u4e0a\u3002\u72d7\u4e3b\u4eba\u8981\u7532\u72d7\u7dca\u76ef\u8457\u4e59\u72d7\u3001\u4e59\u72d7\u7dca\u76ef\u8457\u4e19\u72d7\u3001\u4e19\u72d7\u7dca\u76ef\u8457\u4e01\u72d7\u3001\u4e01\u72d7\u7dca\u76ef\u8457\u7532\u72d7\u3002\u4e00\u8072\u4ee4\u4e0b\uff0c\u56db\u96bb\u72d7\u4ee5\u76f8\u540c\u7684\u901f\u5ea6\u540c\u6642\u885d\u5411\u76ee\u6a19\u3002\u5047\u5b9a\u6bcf\u96bb\u72d7\u5728\u6bcf\u500b\u6642\u523b\u90fd\u662f\u6b63\u9762\u671d\u5411\u5b83\u7684\u76ee\u6a19\uff0c\u90a3\u9ebc\uff0c\u9019\u56db\u96bb\u72d7\u6240\u8dd1\u904e\u7684\u8def\u5f91\u662f\u4ec0\u9ebc\u5f62\u5f0f\u5462\uff1f<!--more--><\/p>\n<p>\u5047\u8a2d\u56db\u96bb\u72d7\u5728\u67d0\u4e00\u6642\u523b\u7684\u4f4d\u7f6e\u5206\u5225\u70ba \\(A_1\\)\u3001\\(B_1\\)\u3001\\(C_1\\)\u3001\\(D_1\\)\uff08\u53c3\u770b\u5716\u4e00\uff09\uff0c\u5247\u6839\u64da\u56db\u96bb\u72d7\u7684\u884c\u52d5\u4e00\u81f4\u6240\u7522\u751f\u7684\u5c0d\u7a31\u6027\uff0c\u53ef\u77e5 \\(A_1B_1C_1D_1\\) \u4e5f\u662f\u6b63\u65b9\u5f62\uff0c\u800c\u4e14\u5b83\u7684\u4e2d\u5fc3\u4e5f\u5c31\u662f\u6b63\u65b9\u5f62 \\(ABCD\\) \u7684\u4e2d\u5fc3 \\(O\\)\u3002<\/p>\n<p>\u66f4\u9032\u4e00\u6b65\u5730\uff0c\u7531\u65bc\u5728 \\(A_1\\) \u9ede\u7684\u7532\u72d7\u4fc2\u885d\u5411\u5728 \\(B_1\\) \u9ede\u7684\u4e59\u72d7\uff0c\u6240\u4ee5\uff0c\u7532\u72d7\u5728\u6b64\u4e00\u6642\u523b\u7684\u901f\u5ea6\u65b9\u5411\u5728\u5411\u91cf \\(\\overrightarrow{A_1B_1}\\) \u4e0a\u3002\u6216\u8005\u8aaa\uff0c\u7532\u72d7\u6240\u8dd1\u7684\u8def\u5f91\u5728 \\(A_1\\) \u9ede\u7684\u5207\u7dda\u8207\u76f4\u7dda \\(OA_1\\) \u5f62\u6210 \\(45^\\circ\\) \u7684\u593e\u89d2\u3002\u540c\u7406\uff0c\u4e59\u72d7\u6240\u8dd1\u7684\u8def\u5f91\u5728 \\(B_1\\) \u9ede\u7684\u5207\u7dda\u8207\u76f4\u7dda \\(OB_1\\) \u5f62\u6210 \\(45^\\circ\\) \u7684\u593e\u89d2\uff0c\u7b49\u7b49\u3002<\/p>\n<div id=\"attachment_39450\" style=\"width: 290px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/wp-content\/uploads\/2011\/09\/On-the-Equiangular-Spiral-1.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-39450\" class=\"size-full wp-image-39450 \" title=\"On the Equiangular Spiral-1\" src=\"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/wp-content\/uploads\/2011\/09\/On-the-Equiangular-Spiral-1.jpg\" alt=\"\" width=\"280\" height=\"255\" \/><\/a><p id=\"caption-attachment-39450\" class=\"wp-caption-text\">\u5716\u4e00<\/p><\/div>\n<p>\u4e00\u822c\u800c\u8a00\uff0c\u82e5\u4e00\u66f2\u7dda\u5728\u6bcf\u500b\u9ede \\(P\\) \u7684\u5207\u5411\u91cf\u90fd\u8207\u67d0\u5b9a\u9ede \\(O\\) \u81f3\u6b64\u9ede \\(P\\) \u6240\u6210\u7684\u5411\u91cf\u593e\u6210\u4e00\u5b9a\u89d2\uff0c\u4e14\u5b9a\u89d2\u4e0d\u662f\u76f4\u89d2\uff0c\u5247\u6b64\u66f2\u7dda\u7a31\u70ba\u4e00\u7b49\u89d2\u87ba\u7dda\uff08equiangular\u00a0spiral\uff09\uff0c\\(O\\) \u9ede\u7a31\u70ba\u5b83\u7684\u6975\u9ede\uff08pole\uff09\u3002\u4e0b\u5716\u4e8c\u662f\u4e00\u7b49\u89d2\u87ba\u7dda\u7684\u90e8\u5206\u5716\u5f62\uff1a<\/p>\n<div id=\"attachment_39451\" style=\"width: 262px\" class=\"wp-caption aligncenter\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-39451\" class=\"size-full wp-image-39451 \" title=\"On the Equiangular Spiral-2\" src=\"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/wp-content\/uploads\/2011\/09\/On-the-Equiangular-Spiral-2.jpg\" alt=\"\" width=\"252\" height=\"230\" \/><p id=\"caption-attachment-39451\" class=\"wp-caption-text\">\u5716\u4e8c<\/p><\/div>\n<p>\u524d\u9762\u6240\u63d0\u7684\u56db\u72d7\u8ffd\u9010\u554f\u984c\u4e2d\uff0c\u6bcf\u96bb\u72d7\u6240\u7d93\u904e\u7684\u8def\u7dda\u90fd\u662f\u4e00\u7b49\u89d2\u87ba\u7dda\u7684\u4e00\u90e8\u5206\uff0c\u6b64\u7b49\u89d2\u87ba\u7dda\u4e2d\u7684\u5b9a\u89d2\u662f \\(\\pi\/4\\)\uff08\u6216 \\(3\\pi\/4\\)\uff0c\u56e0\u70ba\u5207\u5411\u91cf\u53ef\u9078\u6210\u76f8\u53cd\u65b9\u5411\uff09\uff0c\u800c\u5176\u6975\u9ede\u662f\u6b63\u65b9\u5f62 \\(ABCD\\) \u7684\u4e2d\u5fc3 \\(O\\)\u3002<\/p>\n<p>\u5728\u5750\u6a19\u5e73\u9762\u4e0a\uff0c\u82e5\u4e00\u7b49\u89d2\u87ba\u7dda\u7684\u5b9a\u89d2\u70ba \\(\\alpha~(0&lt;\\alpha &lt;\\pi,~\\alpha\\neq 2\\pi)\\)\u3001\u6975\u9ede\u662f\u539f\u9ede \\(O\\)\uff0c\u5247\u6b64\u7b49\u89d2\u87ba\u7dda\u7684\u6975\u5750\u6a19\u65b9\u7a0b\u5f0f\u70ba \\(r=ae^{\\theta \\cos\\alpha}\\)\uff0c\u5176\u4e2d\u7684 \\(a\\) \u662f\u4e00\u5e38\u6578\u3002\u7531\u65bc\u5728\u5c0e\u51fa\u7b49\u89d2\u87ba\u7dda\u6975\u5750\u6a19\u65b9\u7a0b\u5f0f\u7684\u904e\u7a0b\u4e2d\u9700\u8981\u5f15\u7528\u81ea\u7136\u5c0d\u6578\uff0c\u6240\u4ee5\uff0c\u5c0d\u89d2\u87ba\u7dda\u4e5f\u7a31\u70ba\u5c0d\u6578\u87ba\u7dda\uff08logarithmic\u00a0spiral\uff09\u3002<\/p>\n<p><span style=\"color: #000080;\"><strong>\u8da3\u53f2\u4e00\u5247 <\/strong><\/span><\/p>\n<p>\u5c0d\u65bc\u7b49\u89d2\u87ba\u7dda\u7684\u63a2\u8a0e\uff0c\u4ee5\u767d\u52aa\u5229\uff08J. Bernoulli, 1654\uff5e1705\u5e74\uff09\u7684\u6210\u679c\u6700\u70ba\u8c50\u78a9\u3002\u4ed6\u767c\u73fe\u5c07\u7b49\u89d2\u87ba\u7dda\u4f5c\u67d0\u4e9b\u8b8a\u63db\u6642\uff0c\u6240\u5f97\u7684\u66f2\u7dda\u4ecd\u662f\u5168\u7b49\u7684\u5c0d\u89d2\u87ba\u7dda\u3002\u9019\u4e9b\u8b8a\u63db\u5305\u62ec\uff1a\u6c42\u7b49\u89d2\u87ba\u7dda\u7684\u5782\u8db3\u66f2\u7dda\uff08pedal curve\uff09\uff1b\u6c42\u7b49\u89d2\u87ba\u7dda\u7684\u6f38\u5c48\u7dda\uff08evolute\uff09\uff1b\u6c42\u7b49\u89d2\u87ba\u7dda\u7684\u53cd\u6f14\u66f2\u7dda\uff08inversive curve\uff09\uff1b\u6c42\u7b49\u89d2\u87ba\u7dda\u7684\u7126\u7dda\uff08caustic curve\uff09\uff1b\u5c07\u7b49\u89d2\u87ba\u7dda\u4ee5\u5176\u6975\u9ede\u70ba\u4e2d\u5fc3\u4f5c\u4f38\u7e2e\u8b8a\u63db\uff08dilatation\uff09\u3002<\/p>\n<p>\u7531\u65bc\u9019\u4e9b\u8b8a\u63db\u90fd\u53ef\u4ee5\u4f7f\u7b49\u89d2\u87ba\u7dda\u518d\u751f\uff0c\u9019\u500b\u73fe\u8c61\u4f7f\u767d\u52aa\u5229\u5927\u70ba\u6b23\u6170\uff0c\u6240\u4ee5\uff0c\u81e8\u7d42\u524d\u907a\u8a00\u8981\u5c07\u7b49\u89d2\u87ba\u7dda\u7684\u9019\u4e9b\u6027\u8cea\u523b\u5728\u4ed6\u5893\u7891\u4e0a\uff0c\u540c\u6642\u984c\u4e0a\u4e00\u53e5\u8a71\uff1a\u300cEadem mutata resurgo\u300d\uff08\u96d6\u7136\u67d0\u4e9b\u72c0\u6cc1\u6539\u8b8a\u4e86\uff0c\u6211\u537b\u4fdd\u6301\u4e0d\u8b8a\uff09\u3002\u9019\u662f\u7e7c\u963f\u57fa\u7c73\u5fb7\uff08\u7d00\u5143\u524d\u4e09\u4e16\u7d00\uff09\u4e4b\u5f8c\uff0c\u53e6\u4e00\u4f4d\u5728\u5893\u7891\u4e0a\u8868\u73fe\u5176\u6210\u679c\u7684\u6578\u5b78\u5bb6\u3002<\/p>\n<p><span style=\"color: #000080;\"><strong>\u7b49\u89d2\u87ba\u7dda\u7684\u76f8\u4f3c\u6027\u8cea <\/strong><\/span><\/p>\n<p>\u5c0d\u65bc\u4e00\u822c\u7684\u5e7e\u4f55\u5716\u5f62\uff0c\u82e5\u6211\u5011\u9078\u5b9a\u67d0\u500b\u9ede\u505a\u70ba\u4f38\u7e2e\u4e2d\u5fc3\u5c07\u5716\u5f62\u653e\u5927\u6216\u7e2e\u5c0f\uff0c\u5247\u53ef\u5f97\u5230\u4e00\u500b\u76f8\u4f3c\u7684\u5716\u5f62\u3002\u5728\u5c0d\u89d2\u87ba\u7dda\u7684\u60c5\u5f62\u4e2d\uff0c\u82e5\u4f38\u7e2e\u4e2d\u5fc3\u662f\u5b83\u7684\u6975\u9ede\uff0c\u5247\u4e0d\u8ad6\u653e\u5927\u6216\u7e2e\u5c0f\u591a\u5c11\u500d\uff0c\u6240\u5f97\u7684\u4e0d\u53ea\u662f\u76f8\u4f3c\u5716\u5f62\u800c\u5df2\uff0c\u5b83\u662f\u8207\u539f\u7b49\u89d2\u87ba\u7dda\u5168\u7b49\u7684\u4e00\u500b\u7b49\u89d2\u87ba\u7dda\u3002<\/p>\n<p>\u70ba\u4ec0\u9ebc\u5462\uff1f\u82e5\u4ee5\u6975\u9ede\u70ba\u4f38\u7e2e\u4e2d\u5fc3\u5c07\u7b49\u89d2\u87ba\u7dda \\(r=ae^{\\theta \\cos\\alpha}\\) \u4f38\u7e2e \\(m\\) \u500d\uff0c\u5247\u6240\u5f97\u7684\u5716\u5f62\u662f\u7b49\u89d2\u87ba\u7dda \\(r=ame^{\\theta \\cos\\alpha}\\)\u3002\u56e0\u70ba \\(m&gt;0\\)\uff0c\u6240\u4ee5\u53ef\u627e\u5230\u4e00\u500b\u5be6\u6578 \\(\\phi\\) \u4f7f\u5f97 \\(m=e^{\\phi \\cos\\alpha}\\)\u3002\u65bc\u662f\u4f38\u7e2e\u5f8c\u7684\u5716\u5f62\u70ba \\(r=ae^{(\\theta +\\phi) \\cos\\alpha}\\)\uff0c\u9019\u500b\u5716\u5f62\u5176\u5be6\u5c31\u662f\u5c07\u7b49\u89d2\u87ba\u7dda \\(r=ae^{\\theta \\cos\\alpha}\\) \u7e5e\u6975\u9ede\u9806\u6642\u91dd\u65cb\u8f49 \\(\\phi\\) \u89d2\u6240\u5f97\u7684\uff0c\u5b83\u81ea\u7136\u8207\u539f\u7b49\u89d2\u87ba\u7dda \\(r=ae^{\\theta \\cos\\alpha}\\) \u5168\u7b49\u3002<\/p>\n<p>\u6839\u64da\u524d\u6bb5\u7684\u8aaa\u660e\u53ef\u77e5\uff1a\u7b49\u89d2\u87ba\u7dda\u4e0a\u7684\u4e00\u6bb5\u5f27\u7d93\u4f38\u7e2e\u82e5\u5e72\u500d\u5f8c\uff0c\u5fc5\u8207\u8a72\u7b49\u89d2\u87ba\u7dda\u4e0a\u7684\u53e6\u4e00\u6bb5\u5f27\u5168\u7b49\u3002\u4e8b\u5be6\u4e0a\uff0c\u82e5 \\(r=ae^{\\theta \\cos\\alpha}\\) \u7d93\u4f38\u7e2e\u800c\u6210 \\(r=ae^{(\\theta +\\phi) \\cos\\alpha}\\)\uff0c\u5247\u5728\u7b49\u89d2\u87ba\u7dda \\(r=ae^{\\theta \\cos\\alpha}\\) \u4e0a\uff0c\u8f3b\u89d2 \\(\\theta\\) \u6eff\u8db3 \\(\\beta\\leq\\theta\\leq\\gamma\\) \u7684\u5f27\uff0c\u7d93\u4f38\u7e2e\u5f8c\u5fc5\u8207\u8a72\u7b49\u89d2\u87ba\u7dda\u4e0a\u8f3b\u89d2 \\(\\theta\\) \u6eff\u8db3 \\(\\beta+\\phi\\leq\\theta\\leq\\gamma+\\phi\\) \u7684\u5f27\u5168\u7b49\u3002<\/p>\n<p>\u7b49\u89d2\u87ba\u7dda\u7684\u9019\u9805\u7279\u6027\uff0c\u4f7f\u5f97\u81ea\u7136\u754c\u4e2d\u8a31\u591a\u7269\u9ad4\u90fd\u5448\u73fe\u7b49\u89d2\u87ba\u7dda\u7684\u5f62\u72c0\u3002\u4f8b\u5982\uff1a\u8a31\u591a\u8c9d\u6bbc\u7684\u5f62\u72c0\u90fd\u5f88\u63a5\u8fd1\u7b49\u89d2\u87ba\u7dda\uff0c\u56e0\u70ba\u751f\u6d3b\u5728\u6bbc\u5167\u7684\u52d5\u7269\u5728\u6210\u9577\u904e\u7a0b\u4e2d\u901a\u5e38\u662f\u5747\u52fb\u5730\u9577\u5927\uff0c\u9019\u5c31\u50cf\u76f8\u4f3c\u5730\u653e\u5927\uff0c\u6240\u4ee5\uff0c\u65b0\u751f\u7684\u90e8\u5206\u6240\u68f2\u606f\u7684\u7a7a\u9593\u5fc5\u8207\u539f\u6709\u7a7a\u9593\u5f62\u72c0\u76f8\u4f3c\u3002\u8c61\u9f3b\u3001\u52d5\u7269\u7684\u89d2\u8207\u6bdb\u7b49\u90fd\u5448\u7b49\u89d2\u87ba\u7dda\u5f62\u3002\u5728\u690d\u7269\u4e2d\uff0c\u5411\u65e5\u8475\u3001\u9cf3\u68a8\u8207\u96db\u83ca\u4e0a\u7684\u87ba\u65cb\u7d0b\u4e5f\u90fd\u5448\u7b49\u89d2\u87ba\u7dda\u5f62\u3002\u57163\u662f\u9e1a\u9d61\u87ba\u7684\u6a6b\u622a\u9762\uff0c\u9019\u9ebc\u7f8e\u7684\u7dda\u689d\uff0c\u4ee4\u4eba\u4e0d\u5f97\u4e0d\u4f69\u670d\u9020\u7269\u4e4b\u5947\u3002<\/p>\n<div id=\"attachment_39455\" style=\"width: 363px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/wp-content\/uploads\/2011\/09\/On-the-Equiangular-Spiral-3.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-39455\" class=\"size-full wp-image-39455 \" title=\"On the Equiangular Spiral-3\" src=\"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/wp-content\/uploads\/2011\/09\/On-the-Equiangular-Spiral-3.jpg\" alt=\"\" width=\"353\" height=\"258\" \/><\/a><p id=\"caption-attachment-39455\" class=\"wp-caption-text\">\u5716\u4e09<\/p><\/div>\n<p><span style=\"color: #000080;\"><strong>\u9ec3\u91d1\u5206\u5272\u8207\u7b49\u89d2\u87ba\u7dda<\/strong><\/span><\/p>\n<p>\u74b0\u7e5e\u67d0\u500b\u5b9a\u9ede\u800c\u76f8\u4f3c\u5730\u7e2e\u5c0f\uff0c\u9019\u662f\u7b49\u89d2\u87ba\u7dda\u5728\u5176\u6975\u9ede\u9644\u8fd1\u5448\u73fe\u7684\u73fe\u8c61\u3002\u5047\u5982\u6211\u5011\u5c07\u591a\u908a\u5f62\u74b0\u7e5e\u4e00\u5b9a\u9ede\u800c\u76f8\u4f3c\u5730\u7e2e\u5c0f\uff0c\u662f\u4e0d\u662f\u6703\u8207\u7b49\u89d2\u87ba\u7dda\u7522\u751f\u95dc\u806f\u5462\uff1f<\/p>\n<p>\u5728\u5716\u56db\u4e2d\uff0c\\(\\Box ABDF\\)\u3001\\(\\Box CDFH\\)\u3001\\(\\Box EFHJ\\)\u3001\\(\\Box GHJK\\)\u3001\\(\\Box IJKL\\) \u662f\u4e00\u7cfb\u5217\u7684\u77e9\u5f62\uff0c\u9019\u4e9b\u77e9\u5f62\u4e2d\u6bcf\u5169\u500b\u90fd\u76f8\u4f3c\uff08\u4ea6\u5373\uff1a\u908a\u7684\u6bd4\u503c\u76f8\u7b49\uff09\uff0c\u800c\u4e14\u5f8c\u4e00\u500b\u77e9\u5f62\u90fd\u662f\u7531\u5176\u524d\u9762\u7684\u77e9\u5f62\u6316\u6389\u4e00\u500b\u6b63\u65b9\u5f62\u800c\u5f97\u7684\u3002\u5982\uff1a\\(\\Box{CDFH}\\) \u662f\u7531 \\(\\Box ABDF\\) \u6316\u6389\u6b63\u65b9\u5f62 \\(\\Box ABCH\\) \u800c\u5f97\u7684\u3002<\/p>\n<p>\u6b64\u6642\uff0c\u4e0a\u5217\u77e9\u5f62\u7684\u7b2c\u4e00\u500b\u9802\u9ede \\(A\\)\u3001\\(C\\)\u3001\\(E\\)\u3001\\(G\\)\u3001\\(I\\)\u3001\\(K\\) \u7b49\u6703\u843d\u5728\u4e00\u7b49\u89d2\u87ba\u7dda\u4e0a\uff0c\u6b64\u7b49\u89d2\u87ba\u7dda\u7684\u6975\u9ede\u662f\u76f4\u7dda \\(AE\\)\u3001\\(BF\\)\u3001\\(CG\\)\u3001\\(DH\\) \u7b49\u5171\u540c\u7684\u4ea4\u9ede \\(O\\)\u3002<\/p>\n<p>\u82e5\u4ee5 \\(O\\) \u70ba\u6975\u9ede\uff0c\u5c04\u7dda \\(\\overrightarrow{OE}\\) \u70ba\u6975\u8ef8\uff0c\u4e14 \\(A\\) \u7684\u6975\u5750\u6a19\u70ba \\((a,\\pi)\\)\uff0c\u5247\u6b64\u7b49\u89d2\u87ba\u7dda\u7684\u6975\u5750\u6a19\u65b9\u7a0b\u5f0f\u70ba<\/p>\n<p style=\"text-align: center;\">\\(\\displaystyle r=\\frac{a}{\\phi^2}(\\phi^{2\/\\pi})^\\theta\\)<\/p>\n<p>\u5176\u4e2d\uff0c\\(\\phi=(1+\\sqrt{5})\/2\\)\u3002\u9019\u500b\u7b49\u89d2\u87ba\u7dda\u901a\u5e38\u7a31\u70ba\u9ec3\u91d1\u87ba\u7dda\u3002<\/p>\n<p>\u70ba\u4ec0\u9ebc\u6703\u626f\u4e0a \\(\\frac{1+\\sqrt{5}}{2}\\) \u5462\uff1f\u539f\u4f86\u9019\u500b\u6578\u5c31\u662f\u4e0a\u8ff0\u76f8\u4f3c\u77e9\u5f62\u7684\u9577\u908a\u8207\u77ed\u908a\u7684\u9577\u4e4b\u6bd4 \\(\\overline{BC}:\\overline{CD}\\)\u3002<\/p>\n<p>\u82e5\u7dda\u6bb5 \\(\\overline{BD}\\) \u4e0a\u7684\u4e00\u9ede \\(C\\) \u6eff\u8db3 \\(\\overline{BD}:\\overline{BC}=\\overline{BC}:\\overline{CD}\\)\uff0c\u5247\u7a31 \\(C\\) \u9ede\u5c07 \\(\\overline{BD}\\) \u9ec3\u91d1\u5206\u5272\u3002\u7576 \\(C\\) \u9ede\u5c07 \\(\\overline{BD}\\) \u9ec3\u91d1\u5206\u5272\u6642\uff0c\\(\\overline{BC}:\\overline{CD}\\) \u7684\u503c\u662f \\((1+\\sqrt{5})\/2\\)\uff0c\u6b64\u6578\u7a31\u70ba\u9ec3\u91d1\u5206\u5272\u6bd4\u3002\u82e5\u4e00\u77e9\u5f62\u7684\u9577\u908a\u8207\u77ed\u908a\u7684\u6bd4\u503c\u70ba \\((1+\\sqrt{5})\/2\\)\uff0c\u5247\u6b64\u77e9\u5f62\u7a31\u70ba\u9ec3\u91d1\u77e9\u5f62\u3002<\/p>\n<div id=\"attachment_39457\" style=\"width: 344px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/wp-content\/uploads\/2011\/09\/On-the-Equiangular-Spiral-5.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-39457\" class=\"size-full wp-image-39457 \" title=\"On the Equiangular Spiral-5\" src=\"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/wp-content\/uploads\/2011\/09\/On-the-Equiangular-Spiral-5.jpg\" alt=\"\" width=\"334\" height=\"223\" \/><\/a><p id=\"caption-attachment-39457\" class=\"wp-caption-text\">\u5716\u56db<\/p><\/div>\n<p>\u7531\u9ec3\u91d1\u77e9\u5f62\u53ef\u5f15\u51fa\u7b49\u89d2\u87ba\u7dda\uff0c\u5c07\u77e9\u5f62\u6539\u6210\u4e09\u89d2\u5f62\uff0c\u4e5f\u6703\u6709\u540c\u6a23\u7684\u7d50\u679c\u55ce\uff1f<\/p>\n<p>\u5728\u5716\u4e94\u4e2d\uff0c\\(\\Delta ABC\\)\u3001\\(\\Delta BCD\\)\u3001\\(\\Delta CDE\\)\u3001\\(\\Delta DEF\\)\u3001\\(\\Delta EFG\\)\u3001\\(\\Delta FGH\\) \u7b49\u662f\u4e00\u7cfb\u5217\u7684\u7b49\u8170\u4e09\u89d2\u5f62\uff0c\u9019\u4e9b\u7b49\u8170\u4e09\u89d2\u5f62\u4e2d\u6bcf\u5169\u500b\u90fd\u76f8\u4f3c\uff0c\u800c\u4e14\u5f8c\u4e00\u7b49\u8170\u4e09\u89d2\u5f62\uff0c\u90fd\u662f\u7531\u5176\u524d\u9762\u7684\u7b49\u8170\u4e09\u89d2\u5f62\u6316\u6389\u4e00\u500b\u7b49\u8170\u4e09\u89d2\u5f62\u800c\u5f97\u7684\u3002<\/p>\n<p>\u4f8b\u5982\uff1a\\(\\Delta BCD\\) \u662f\u7531 \\(\\Delta ABC\\) \u6316\u6389\u7b49\u8170\u4e09\u89d2\u5f62 \\(\\Delta DAB\\) \u800c\u5f97\u7684\u3002\u6b64\u6642\uff0c\u4e0a\u5217\u7b49\u8170\u4e09\u89d2\u5f62\u7684\u9802\u9ede \\(A\\)\u3001\\(B\\)\u3001\\(C\\)\u3001\\(D\\)\u3001\\(E\\)\u3001\\(F\\)\u3001\\(G\\)\u3001\\(H\\)\u3001\\(\\cdots\\)\u7b49\u6703\u843d\u5728\u4e00\u7b49\u89d2\u87ba\u7dda\u4e0a\uff0c\u6b64\u7b49\u89d2\u87ba\u7dda\u7684\u6975\u9ede\u662f \\(AF\\) \u8207 \\(BG\\) \u7684\u4ea4\u9ede \\(O\\)\u3002<\/p>\n<p>\u82e5\u4ee5 \\(O\\) \u70ba\u6975\u9ede\u3001\u5c04\u7dda \\(OB\\) \u70ba\u6975\u8ef8\u3001\u4e14 \\(A\\) \u7684\u6975\u5750\u6a19\u70ba \\((a,\\frac{3\\pi}{5})\\)\uff0c\u5247\u6b64\u7b49\u89d2\u87ba\u7dda\u7684\u6975\u5750\u6a19\u65b9\u7a0b\u5f0f\u70ba<\/p>\n<p style=\"text-align: center;\">\\(\\displaystyle r=\\frac{a}{\\phi}(\\phi^{5\/(3\\pi)})^\\theta\\)<\/p>\n<p>\u5176\u4e2d\uff0c\\(\\phi=(1+\\sqrt{5})\/2\\)\u3002\u9019\u500b\u7b49\u89d2\u87ba\u7dda\u4e5f\u7a31\u70ba\u9ec3\u91d1\u87ba\u7dda\u3002<\/p>\n<p>\u6b64\u7b49\u89d2\u87ba\u7dda\u4e5f\u626f\u4e0a \\((1+\\sqrt{5})\/2\\)\uff0c\u5176\u7406\u7531\u5982\u4e0b\uff1a\u4e0a\u8ff0\u7684\u76f8\u4f3c\u7b49\u8170\u4e09\u89d2\u5f62 \\(\\Delta ABC\\) \u7b49\uff0c\u53ef\u8b49\u660e\u5176\u9802\u89d2\u70ba \\(36^\\circ\\)\u3001\u5e95\u89d2\u70ba \\(72^\\circ\\)\uff0c\u6240\u4ee5\uff0c\\(\\overline{AB}:\\overline{BC}=(1+\\sqrt{5})\/2\\)\u3002\u6b64\u7a2e\u4e09\u89d2\u5f62\u7a31\u70ba\u9ec3\u91d1\u4e09\u89d2\u5f62\u3002<\/p>\n<div id=\"attachment_39462\" style=\"width: 303px\" class=\"wp-caption aligncenter\"><a href=\"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/wp-content\/uploads\/2011\/09\/On-the-Equiangular-Spiral-7.jpg\"><img loading=\"lazy\" decoding=\"async\" aria-describedby=\"caption-attachment-39462\" class=\"size-full wp-image-39462 \" title=\"On the Equiangular Spiral-7\" src=\"http:\/\/highscope.ch.ntu.edu.tw\/wordpress\/wp-content\/uploads\/2011\/09\/On-the-Equiangular-Spiral-7.jpg\" alt=\"\" width=\"293\" height=\"358\" \/><\/a><p id=\"caption-attachment-39462\" class=\"wp-caption-text\">\u5716\u4e94<\/p><\/div>\n<p><strong>\u5ef6\u4f38\u95b1\u8b80\uff1a<\/strong><\/p>\n<ol>\n<li><a href=\"http:\/\/episte.math.ntu.edu.tw\/cgi\/mathfield.pl?aut=%BB%AF%A4%E5%B1%D3\" target=\"_blank\" rel=\"noopener noreferrer\">http:\/\/episte.math.ntu.edu.tw\/cgi\/mathfield.pl?aut=%BB%AF%A4%E5%B1%D3<\/a><\/li>\n<\/ol>\n<\/div>","protected":false},"excerpt":{"rendered":"<p>\u8a8d\u8b58\u7b49\u89d2\u87ba\u7dda(On the Equiangular Spiral) \u570b\u7acb\u81fa\u7063\u5e2b\u7bc4\u5927\u5b78\u6578\u5b78\u7cfb\u8d99\u6587\u654f\u6559\u6388\/\u570b\u7acb\u81fa\u7063&hellip;<\/p>\n","protected":false},"author":50,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[224,111],"tags":[4353,4354,4311],"class_list":["post-39449","post","type-post","status-publish","format-standard","hentry","category-math04","category-mathematics00","tag-4353","tag-4354","tag-4311","loop-entry","cat-224","cat-111","no-thumbnail"],"views":10872,"_links":{"self":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/39449","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/users\/50"}],"replies":[{"embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/comments?post=39449"}],"version-history":[{"count":1,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/39449\/revisions"}],"predecessor-version":[{"id":87844,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/posts\/39449\/revisions\/87844"}],"wp:attachment":[{"href":"http:\/\/localhost\/wp-json\/wp\/v2\/media?parent=39449"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/categories?post=39449"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/localhost\/wp-json\/wp\/v2\/tags?post=39449"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}