{"id":3424,"date":"2024-08-18T22:48:06","date_gmt":"2024-08-18T14:48:06","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=3424"},"modified":"2024-08-20T17:46:08","modified_gmt":"2024-08-20T09:46:08","slug":"introduction_computational_geometry_topology__applications_game_worlds","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2024\/08\/18\/introduction_computational_geometry_topology__applications_game_worlds\/","title":{"rendered":"\u6e38\u620f\u4e16\u754c\u7684\u8ba1\u7b97\u51e0\u4f55\u3001\u62d3\u6251\u53ca\u5176\u5e94\u7528\u524d\u8a00"},"content":{"rendered":"<p><script type=\"text\/javascript\" async src=\"https:\/\/www.caiqinyi.cn\/wp-content\/MathJax\/MathJax.js?config=TeX-AMS_CHTML\">\n<\/script><br \/>\n<script type=\"text\/x-mathjax-config\">\n    MathJax.Hub.Config({\n        tex2jax: {inlineMath: [['$','$']]},\n        TeX: {equationNumbers: {autoNumber: [\"AMS\"], useLabelIds: true}},\n        \"HTML-CSS\": {linebreaks: {automatic: true}},\n        SVG: {linebreaks: {automatic: true}}\n    });\n<\/script><\/p>\n<p>\u4eca\u5e74\u5e74\u521d\u5f00\u59cb, \u6211\u7684\u7814\u7a76\u91cd\u5fc3\u4fbf\u4ece\u5bf9\u4e8e\u5355\u4e2a\u6e38\u620f\u6a21\u578b\u7684\u62d3\u6251\u5206\u6790\u8f6c\u79fb\u5230\u5bf9\u4e8e\u6574\u4e2a\u6e38\u620f\u4e16\u754c\u7684\u62d3\u6251\u5206\u6790. \u524d\u6bb5\u65f6\u95f4\u5076\u7136\u6dd8\u5230\u4e00\u672c\u4e66\u300a\u6570\u5b57\u56fe\u50cf\u7684\u8ba1\u7b97\u51e0\u4f55\u3001\u62d3\u6251\u548c\u7269\u7406\u53ca\u5176\u5e94\u7528\u300b, \u4f5c\u8005\u662f\u6765\u81ea\u52a0\u62ff\u5927\u9a6c\u5c3c\u6258\u5df4\u5927\u5b66\u7684James Peters\u6559\u6388, \u5176Erdos\u6570\u4e3a3, \u662f\u4e00\u4f4d\u633a\u5f3a\u7684\u7814\u7a76\u8ba1\u7b97\u62d3\u6251\u7684\u5b66\u8005. \u51a5\u51a5\u4e2d\u89c9\u5f97\u80fd\u4ece\u8be5\u4e66\u4e2d\u83b7\u53d6\u5982\u4f55\u53bb\u7814\u7a76\u8ba1\u7b97\u62d3\u6251\u5728\u6e38\u620f\u4e16\u754c\u4e2d\u7684\u5e94\u7528\u7684\u7075\u611f, \u76ee\u524d\u4e5f\u5df2\u7ecf\u7c97\u7565\u9605\u8bfb\u4e86\u4e09\u5206\u4e4b\u4e8c\u7684\u5185\u5bb9\u4e86, \u4f46\u6536\u83b7\u4e0d\u7b97\u592a\u591a, \u8ba9\u6211\u6709\u4e9b\u5931\u671b\u7684\u70b9\u4e3b\u8981\u5728\u4e8e\u5927\u90e8\u5206\u5b9a\u7406\u7684\u8bc1\u660e\u6709\u70b9\u5faa\u73af\u8bba\u8bc1\u7684\u5473\u9053. \u5c3d\u7ba1\u5982\u6b64, \u6211\u4f9d\u65e7\u60f3\u8bb0\u5f55\u4e0b\u8bfb\u7f62\u8be5\u4e66\u7684\u4e00\u4e9b\u6536\u83b7, \u672c\u6587\u4fbf\u662f\u4e00\u7bc7\u5173\u4e8e\u6e38\u620f\u4e16\u754c\u7684\u8ba1\u7b97\u51e0\u4f55\u3001\u62d3\u6251\u53ca\u5176\u5e94\u7528\u7684\u524d\u8a00.<\/p>\n<p><!--more--><\/p>\n<p>Edelsbrunner\u7684\u7f51\u683c\u751f\u6210\u51e0\u4f55\u548cZiegler\u7684\u591a\u80de\u5f62\u51e0\u4f55\u4e3a\u6211\u4eec\u6240\u7814\u7a76\u7684\u4e09\u89d2\u5316\u573a\u666f\u7684\u51e0\u4f55\u7ed3\u6784\u63d0\u4f9b\u4e86\u8ba1\u7b97\u51e0\u4f55\u5b66\u65b9\u6cd5\u7684\u575a\u5b9e\u57fa\u7840. \u4e09\u7ef4\u591a\u80de\u5f62\u7531\u8986\u76d6\u6e38\u620f\u6a21\u578b\u5185\u90e8\u7684\u95ed\u5408\u534a\u7a7a\u95f4\u76f8\u4ea4\u6240\u5b9a\u4e49\u7684\u586b\u5145\u591a\u9762\u4f53. \u7531Edelsbrunner\u548cHarer\u63d0\u51fa\u7684\u5355\u5143\u590d\u5408\u578b\u62d3\u6251\u7ed3\u6784\u4ee5\u53ca\u7531Munch\u5728\u6301\u4e45\u6027\u540c\u6e90\u6027\u65b9\u9762\u7684\u5de5\u4f5c\u4e3a\u6e38\u620f\u573a\u666f\u7684\u8ba1\u7b97\u62d3\u6251\u5b66\u7684\u4ecb\u7ecd\u6027\u7814\u7a76\u63d0\u4f9b\u4e86\u575a\u5b9e\u7684\u57fa\u7840. \u8fd9\u79cd\u62d3\u6251\u5f62\u5f0f\u63a2\u7d22\u4e86\u6e38\u620f\u573a\u666f\u4e2d\u5e38\u89c1\u7684\u6784\u9020\u3001\u5f62\u72b6\u548c\u7ed3\u6784.<br \/>\n$\\\\$ \u7814\u7a76\u6e38\u620f\u6a21\u578b\u76ee\u6807\u5f62\u72b6\u5728\u6e38\u620f\u5e27\u5e8f\u5217\u4ee5\u53ca\u8bb0\u5f55\u6e38\u620f\u573a\u666f\u4e2d\u8868\u9762\u5f62\u72b6\u53d8\u5316\u7684\u5e27\u5e8f\u5217\u4e2d\u7684\u6301\u4e45\u6027\u975e\u5e38\u91cd\u8981. \u8868\u9762\u5f62\u72b6\u4f1a\u51fa\u73b0\u51fa\u73b0\u3001\u7ecf\u5386\u5404\u79cd\u5149\u7ebf\u548c\u8868\u9762\u6761\u4ef6\u7684\u53d8\u5316\u3001\u5e76\u6700\u7ec8\u6d88\u5931. \u73a9\u5bb6\u901a\u5e38\u503e\u5411\u4e8e\u5728\u6e38\u620f\u573a\u666f\u4e2d\u5bfb\u627e\u4e0d\u5bfb\u5e38\u7684\u7269\u4f53(\u81ea\u7136\u7684\u548c\u4eba\u5de5\u7684), \u8fd9\u662f\u4e00\u79cd\u5bf9\u6e38\u620f\u573a\u666f\u4e2d\u8fde\u7eed\u53d8\u5316\u548c\u89c2\u5bdf\u5230\u7684\u6210\u5206\u7684\u77ac\u95f4\u6301\u4e45\u6027\u7684\u9ed8\u8ba4\u8ba4\u8bc6. \u6362\u53e5\u8bdd\u8bf4, \u91cd\u8981\u7684\u662f\u9700\u8003\u8651\u6e38\u620f\u573a\u666f\u5f62\u72b6\u7684\u65f6\u7a7a\u7279\u6027. \u8fd9\u6307\u5bf9\u6e38\u620f\u573a\u666f\u7684\u7406\u89e3\u5305\u62ec\u5bf9\u6e38\u620f\u573a\u666f\u7684\u51e0\u4f55\u548c\u62d3\u6251\u7684\u7814\u7a76.<br \/>\n$\\\\$ \u8ba1\u7b97\u51e0\u4f55\u5b66\u6709\u52a9\u4e8e\u6355\u83b7\u5d4c\u5165\u5728\u6e38\u620f\u6a21\u578b\u76ee\u6807\u5f62\u72b6\u4e2d\u7684\u7ec6\u7c92\u5ea6\u7ed3\u6784. \u8ba1\u7b97\u62d3\u6251\u5b66\u80fd\u591f\u6355\u6349\u548c\u5206\u6790\u5d4c\u5165\u5728\u4e09\u89d2\u5316\u6e38\u620f\u573a\u666f\u51e0\u4f55\u7ed3\u6784\u4e2d\u7684\u5355\u5143\u590d\u5408\u5f62(\u9876\u70b9\u3001\u7ebf\u6bb5\u3001\u586b\u5145\u4e09\u89d2\u5f62\u3001\u5faa\u73af\u3001\u6da1\u65cb\u3001\u795e\u7ecf\u7684\u96c6\u65cf) \u91cc\u53d1\u73b0\u7684\u8fd1\u90bb. \u5355\u5143\u590d\u5408\u5f62\u7684\u540c\u6e90\u6027(\u4e9a\u5386\u5c71\u5fb7\u7f57\u592b\u62d3\u6251\u5b66\u65b9\u6cd5\u7684\u540e\u4ee3) \u5728\u8fd9\u91cc\u662f\u4e00\u4e2a\u91cd\u8981\u7684\u7ec4\u6210\u90e8\u5206. \u540c\u6e90\u6027\u662f\u4e00\u4e2a\u6570\u5b66\u6846\u67b6, \u5b83\u5173\u6ce8\u7a7a\u95f4\u662f\u5982\u4f55\u8fde\u901a\u7684, \u5e76\u5229\u7528\u4ee3\u6570\u7ed3\u6784, \u5982\u7fa4\u548c\u6620\u5c04, \u5c06\u7a7a\u95f4\u4e2d\u5177\u6709\u62d3\u6251\u610f\u4e49\u7684\u5b50\u96c6\u76f8\u4e92\u8054\u7cfb\u8d77\u6765.<br \/>\n$\\\\$ \u5b9e\u9645\u4e0a, \u540c\u6e90\u6027\u662f\u6d1e\u5bdf\u6e38\u620f\u573a\u666f\u4e2d\u7684\u5404\u4e2a\u90e8\u5206\u5982\u4f55\u76f8\u4e92\u8fde\u901a\u7684\u6765\u6e90. \u5faa\u73af\u7fa4\u6709\u52a9\u4e8e\u4ee5\u7b80\u6d01\u7684\u65b9\u5f0f\u8868\u793a\u6e38\u620f\u573a\u666f\u4e2d\u7684\u5404\u4e2a\u90e8\u5206\u662f\u5982\u4f55\u76f8\u4e92\u8fde\u901a\u5e76\u63a5\u5408\u5728\u4e00\u8d77\u7684. \u5177\u6709\u591a\u4e2a\u751f\u6210\u5143\u7684\u5faa\u73af\u7fa4\u4e5f\u662f\u6307\u5b9a\u611f\u5174\u8da3\u8868\u9762\u5f62\u72b6\u7684\u91cd\u8981\u7279\u5f81\u7684\u4e00\u4e2a\u6765\u6e90, \u8be5\u7279\u5f81\u5373\u8d1d\u8482\u6570(Betti number, \u81ea\u7531\u963f\u8d1d\u5c14\u7fa4\u4e2d\u751f\u6210\u5143\u6570\u91cf\u7684\u8ba1\u6570). H. Poincare\u4e3a\u7eaa\u5ff5Enrico Betti\u547d\u540d\u4e86\u8fd9\u4e2a\u6570. \u6e38\u620f\u4e16\u754c\u7684\u8ba1\u7b97\u51e0\u4f55\u5b66\u3001\u62d3\u6251\u5b66\u7684\u91cd\u70b9\u662f\u6709\u9650\u7a7a\u95f4.<br \/>\n$\\\\$ \u8d1d\u8482\u89c2\u5bdf\u5230, \u6709\u9650\u7a7a\u95f4\u5177\u6709\u4e0e\u5176\u7ef4\u5ea6\u5927\u5c0f\u548c\u5143\u7d20\u5f62\u72b6\u65e0\u5173\u7684\u5c5e\u6027. \u8fd9\u4e9b\u5c5e\u6027\u4ec5\u6307\u5176\u5404\u90e8\u5206\u7684\u8fde\u901a\u65b9\u5f0f\u2026\u2026 \u6709\u9650\u6709\u754c\u7a7a\u95f4\u533a\u57df\u7684\u7279\u6027\u503e\u5411\u4e8e\u901a\u8fc7\u8986\u76d6\u7a7a\u95f4\u7684\u5355\u5143\u590d\u5408\u5f62\u4e2d\u7684\u901a\u9053\u6240\u8fde\u901a\u7684\u9876\u70b9\u6765\u63ed\u793a.<br \/>\n$\\\\$ Kaczynski\u3001Mischaikov\u548cMrozek\u5bf9\u540c\u6e90\u6027\u7684\u8ba1\u7b97\u65b9\u6cd5\u8fdb\u884c\u4e86\u6df1\u5165\u7814\u7a76. \u8fd9\u91cc\u7684\u91cd\u70b9\u662f\u8fa8\u522b\u548c\u8ddf\u8e2a\u3001\u5206\u6790\u548c\u8868\u8fbe\u4ee5\u53ca\u8fd1\u4f3c\u79fb\u52a8\u8868\u9762\u5f62\u72b6\u7684\u63a5\u8fd1\u6027. \u4e3a\u4e86\u5e94\u5bf9\u4e00\u4e2a\u6e38\u620f\u573a\u666f\u4e2d(\u4ece\u4e00\u4e2a\u6e38\u620f\u5e27\u5230\u53e6\u4e00\u4e2a\u6e38\u620f\u5e27) \u7684\u8fde\u7eed\u5f62\u72b6\u53d8\u5316, \u63cf\u8ff0\u6027\u90bb\u8fd1\u7a7a\u95f4\u4e0a\u7684\u7279\u5f81\u77e2\u91cf\u4e3a\u6211\u4eec\u63d0\u4f9b\u4e86\u4e00\u79cd\u8868\u793a\u5f62\u72b6\u53d8\u5316\u7684\u65b9\u6cd5, \u8fd9\u4e9b\u53d8\u5316\u8981\u4e48\u5f7c\u6b64\u9760\u8fd1, \u8981\u4e48\u6709\u65f6\u76f8\u8ddd\u5f88\u8fdc.<br \/>\n$\\\\$ \u5728\u6b27\u6c0f\u7a7a\u95f4\u4e2d, \u51e0\u4f55\u7ed3\u6784\u5305\u62ec\u9876\u70b9\u3001\u7ebf\u6bb5\u3001\u5b9e\u5fc3\u4e09\u89d2\u5f62(\u4e09\u8fb9\u591a\u9762\u4f53) \u548c\u7a7a\u5fc3\u56db\u9762\u4f53. \u4e09\u7ef4\u591a\u80de\u5f62\u662f\u5c01\u95ed\u534a\u7a7a\u95f4\u7684\u4ea4\u96c6. \u5355\u4e2a\u4e09\u7ef4\u591a\u80de\u5f62\u662f\u4e00\u4e2a\u7a7a\u95f4\u533a\u57df, \u5185\u90e8\u5145\u6ee1, \u56db\u5468\u90fd\u6709\u8fb9\u754c. \u5728\u62d3\u6251\u8bbe\u7f6e\u4e2d, \u91cd\u70b9\u662f\u5c06\u6e38\u620f\u573a\u666f\u533a\u57df\u5206\u89e3\u4e3a\u975e\u5e38\u7b80\u5355\u7684\u4e09\u7ef4\u591a\u80de\u5f62, \u4f8b\u5982\u6613\u4e8e\u6d4b\u91cf\u548c\u5206\u6790\u7684\u7a7a\u5fc3\u56db\u9762\u4f53. \u8fd9\u79cd\u62d3\u6251\u7684\u57fa\u672c\u6210\u5206\u662f\u5355\u7eaf\u590d\u5408\u5f62\u3001\u5f62\u72b6\u7406\u8bba\u548c\u6301\u4e45\u540c\u6e90\u6027.<br \/>\n$\\\\$ \u8fd9\u9879\u5de5\u4f5c\u80cc\u540e\u7684\u79d8\u5bc6\u662f\u5c06\u5c01\u95ed\u7684\u6e38\u620f\u573a\u666f\u533a\u57df\u5206\u89e3\u6210\u4e00\u7ec4\u5f62\u72b6\u590d\u5408\u5f62, \u8fd9\u4e9b\u5f62\u72b6\u590d\u5408\u5f62\u4e3a\u5f62\u72b6\u5206\u6790\u63d0\u4f9b\u4e86\u57fa\u7840. \u5f62\u72b6\u590d\u5408\u5f62\u7528\u5d4c\u5957\u7684, \u901a\u5e38\u4e0d\u91cd\u53e0\u7684\u6da1\u65cb\u96c6\u65cf\u5bf9\u5f62\u72b6\u8fdb\u884c\u8986\u76d6. \u5728\u6bcf\u4e2a\u590d\u5408\u5f62\u4e2d\u6784\u6210\u795e\u7ecf\u7ed3\u6784\u7684\u4e09\u89d2\u5f62\u96c6\u65cf\u5177\u6709\u5171\u540c\u7684\u9876\u70b9. \u4e00\u822c\u6765\u8bf4, \u5f62\u72b6\u5206\u6790\u7684\u76ee\u6807\u662f\u5206\u7c7b\u3001\u6bd4\u8f83\u3001\u91cf\u5316\u5f02\u540c, \u5e76\u6d4b\u91cf\u5f62\u72b6\u4e4b\u95f4\u7684\u8ddd\u79bb. \u6211\u4eec\u7684\u7814\u7a76\u91cd\u70b9\u662f\u8ba1\u7b97\u62d3\u6251\u5728\u6e38\u620f\u573a\u666f\u5f62\u72b6\u5206\u6790\u4e2d\u7684\u5e94\u7528.<br \/>\n$\\\\$ \u5bf9\u4e8e\u4e09\u89d2\u5316\u6e38\u620f\u573a\u666f\u4e2d\u7684\u5355\u5143\u590d\u5408\u5f62, \u573a\u666f\u5f62\u72b6\u88ab\u79f0\u4e3a\u795e\u7ecf\u590d\u5408\u5f62\u7684\u586b\u5145\u4e09\u89d2\u5f62\u7c07\u6240\u8986\u76d6. \u8bbe$K$\u662f\u70b9\u96c6\u7684\u6709\u9650\u96c6\u65cf. \u96c6\u65cf$K$\u7684\u795e\u7ecf(\u7528Nrv $K$\u8868\u793a) \u7531$K$\u7684\u6240\u6709\u975e\u7a7a\u5b50\u96c6\u65cf\u7ec4\u6210, \u8fd9\u4e9b\u5b50\u96c6\u65cf\u5177\u6709\u975e\u7a7a\u4ea4\u96c6. \u6bcf\u6761\u795e\u7ecf\u90fd\u6709\u5176\u72ec\u7279\u7684\u5f62\u72b6. \u5728\u4e09\u89d2\u5316\u8868\u9762\u4e0a, \u4e9a\u5386\u5c71\u5fb7\u7f57\u592b\u795e\u7ecf\u590d\u5408\u5f62$A$(\u7528Nrv $A$\u8868\u793a) \u662f\u5177\u6709\u516c\u5171\u9876\u70b9\u7684\u4e09\u89d2\u5f62\u96c6\u65cf.<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u4eca\u5e74\u5e74\u521d\u5f00\u59cb, \u6211\u7684\u7814\u7a76\u91cd\u5fc3\u4fbf\u4ece\u5bf9\u4e8e\u5355\u4e2a\u6e38\u620f\u6a21\u578b\u7684\u62d3\u6251\u5206\u6790\u8f6c\u79fb\u5230\u5bf9\u4e8e\u6574\u4e2a\u6e38\u620f\u4e16\u754c\u7684\u62d3\u6251\u5206\u6790. \u524d\u6bb5\u65f6\u95f4\u5076\u7136\u6dd8\u5230 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2024\/08\/18\/introduction_computational_geometry_topology__applications_game_worlds\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u6e38\u620f\u4e16\u754c\u7684\u8ba1\u7b97\u51e0\u4f55\u3001\u62d3\u6251\u53ca\u5176\u5e94\u7528\u524d\u8a00<\/span><\/a><\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[15,23,24,22],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3424"}],"collection":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/comments?post=3424"}],"version-history":[{"count":12,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3424\/revisions"}],"predecessor-version":[{"id":3434,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3424\/revisions\/3434"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=3424"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=3424"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=3424"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}