{"id":1836,"date":"2022-06-12T21:47:19","date_gmt":"2022-06-12T13:47:19","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=1836"},"modified":"2025-02-26T11:17:08","modified_gmt":"2025-02-26T03:17:08","slug":"hyperbolic_tree","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2022\/06\/12\/hyperbolic_tree\/","title":{"rendered":"\u53cc\u66f2\u6811"},"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>\u5468\u4e94\u53bb\u533b\u9662\u505a\u4e86\u4e00\u6b21\u4f53\u68c0\u590d\u67e5, \u63a5\u4e0b\u6765\u5c31\u8981\u5f00\u542f\u4e00\u4e2a\u6708\u7684\u55d1\u836f\u5386\u7a0b\u4e86. \u54ce, \u5e0c\u671b\u771f\u7684\u53ea\u55d1\u4e00\u4e2a\u6708\u53ed, \u8eab\u4f53\u662f\u9769\u547d\u7684\u672c\u94b1, \u5bf9\u81ea\u5df1\u597d\u70b9~ \u672c\u6587\u7b97\u662f\u7fa4\u4f5c\u7528\u4e00\u8282\u7684\u756a\u5916\u7bc7, \u4ecb\u7ecd\u4e00\u4e0b\u53cc\u66f2\u51e0\u4f55\u5728\u8ba1\u7b97\u673a\u53ef\u89c6\u5316\u9886\u57df\u7684\u4e00\u4e2a\u5e94\u7528\u2014\u2014\u53cc\u66f2\u6811, \u6216\u8bb8\u4e5f\u80fd\u4ece\u4e2d\u5f97\u5230\u7075\u611f.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/06\/A-Focus-Context-Technique-Based-on-Hyperbolic-Geometry-for-Visualizing-Large-Hierarchies.pdf\">A Focus Context Technique Based on Hyperbolic Geometry for Visualizing Large Hierarchies<\/a><br \/>\n2. <a href=\"https:\/\/zh.m.wikipedia.org\/zh-hans\/%E5%87%AF%E8%8E%B1-%E5%85%8B%E8%8E%B1%E5%9B%A0%E6%A8%A1%E5%9E%8B\">\u51ef\u83b1-\u514b\u83b1\u56e0\u6a21\u578b<\/a><\/p>\n<p>\u7403\u9762, \u6b27\u51e0\u91cc\u5f97\u5e73\u9762\u4e0e\u53cc\u66f2\u5e73\u9762\u662f\u4e09\u79cd\u5e38\u89c1\u7684\u51e0\u4f55\u6a21\u578b, \u4e0b\u56fe\u603b\u7ed3\u4e86\u8fd9\u4e09\u79cd\u51e0\u4f55\u6a21\u578b\u7684\u7279\u70b9.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/06\/A-comparison-of-2-dimensional-geometries.png\" alt=\"\" width=\"584\" height=\"741\" class=\"aligncenter size-full wp-image-1839\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/06\/A-comparison-of-2-dimensional-geometries.png 584w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2022\/06\/A-comparison-of-2-dimensional-geometries-236x300.png 236w\" sizes=\"(max-width: 584px) 100vw, 584px\" \/><\/p>\n<p><strong>1. \u53cc\u66f2\u5e73\u9762<\/strong><\/p>\n<p>\u901a\u5e38\u6211\u4eec\u628a\u53cc\u66f2\u5e73\u9762\u7406\u89e3\u6210\u4e0d\u542b\u8fb9\u754c\u7684\u5355\u4f4d\u5706\u76d8$$H^2 = \\{ z \\in C \\ | \\ |z| < 1 \\}$$\u5e26\u4e0a\u53cc\u66f2\u5ea6\u91cf$$d(z_1, z_2) = cosh^{-1}(1 + \\frac{2|z_1 - z_2|^2}{(1 - |z_1|^2)(1 - |z_2|^2))},$$\u5176\u4e2d$cosh^{-1}$\u8868\u793a\u53cd\u53cc\u66f2\u4f59\u5f26, \u5373\u4efb\u53d6\u5b9e\u6570$t > 1$,$$cosh^{-1}(t) = ln(t + \\sqrt{t^2 &#8211; 1}).$$\u8fd9\u4e2a\u7a7a\u95f4\u79f0\u4e3a\u53cc\u66f2\u5e73\u9762\u7684Poincare\u5706\u76d8\u6a21\u578b, \u4e0d\u8fc7\u63d0\u51fa\u8fd9\u4e2a\u6a21\u578b\u7684\u5374\u4e0d\u662fPoincare\u800c\u662fBeltrami. \u53cc\u66f2\u5ea6\u91cf\u7684\u516c\u5f0f\u770b\u4e0a\u53bb\u5f88\u590d\u6742, \u5bf9\u6211\u4eec\u5efa\u7acb\u51e0\u4f55\u76f4\u89c2\u597d\u50cf\u4e0d\u662f\u4ec0\u4e48\u597d\u4e8b\u60c5, \u4f46\u662f\u53cc\u66f2\u7b49\u8ddd\u53d8\u6362\u5374\u975e\u5e38\u5bb9\u6613\u523b\u753b. \u4e00\u4e2a\u81ea\u540c\u80da$f: H^2 \\to $$ H^2$\u5982\u679c\u6ee1\u8db3$d(p, q) = d(f(p), f(q))$, \u5219\u79f0\u4e4b\u4e3a\u53cc\u66f2\u7b49\u8ddd\u53d8\u6362. \u53ef\u4ee5\u8bc1\u660e, \u4efb\u4f55\u4e00\u4e2a\u53cc\u66f2\u7b49\u8ddd\u53d8\u6362\u90fd\u5177\u6709$z \\mapsto $$ \\frac{az + \\bar{b}}{bz + \\bar{a}}$\u6216\u8005$z \\mapsto \\frac{a\\bar{z} + \\bar{b}}{b\\bar{z} + \\bar{a}}$\u7684\u5f62\u5f0f. \u5355\u4f4d\u5706\u76d8$H^2$\u7684\u76f4\u5f84\u4ee5\u53ca\u4e0e\u8fb9\u754c\u5706\u5468\u5782\u76f4\u7684\u5706\u5f27\u7edf\u79f0\u4e3a\u53cc\u66f2\u76f4\u7ebf, \u53cc\u66f2\u7b49\u8ddd\u53d8\u6362\u4e00\u5b9a\u4f1a\u628a\u53cc\u66f2\u76f4\u7ebf\u53d8\u6210\u53cc\u66f2\u76f4\u7ebf, \u5e76\u4e14\u4fdd\u6301\u4e24\u6761\u76f8\u4ea4\u53cc\u66f2\u76f4\u7ebf\u7684\u5939\u89d2\u4e0d\u53d8.<\/p>\n<p><strong>2. \u53cc\u66f2\u6811\u7684\u5b9e\u73b0<\/strong><\/p>\n<p>\u53cc\u66f2\u6811\u7684\u4f7f\u7528\u53ef\u53c2\u8003\u89c6\u9891<a href=\"https:\/\/www.youtube.com\/watch?v=J0yFd2Pq_rg\"> 2:08   2:09 \/ 3:14   Visualizing Large Trees Using the Hyperbolic Browser<\/a>, \u5176\u5e95\u5c42\u5b9e\u73b0\u662f\u4ee5Poincare\u5706\u76d8\u6a21\u578b\u4e3a\u7406\u8bba\u57fa\u7840\u7684. \u6211\u4eec\u7528\u5706\u5fc3\u4f4d\u4e8e\u539f\u70b9\u7684\u5355\u4f4d\u5706\u76d8\u4e0a\u7684\u70b9\u6765\u8868\u793a\u53cc\u66f2\u7a7a\u95f4\u4e2d\u7684\u4e00\u4e2a\u70b9, \u6b64\u65f6\u53cc\u66f2\u5e73\u9762\u4e0a\u7684\u7684\u521a\u6027\u53d8\u6362\u88ab\u8f6c\u6362\u4e3a\u5355\u4f4d\u5706\u76d8\u4e0a\u7684\u4fdd\u5706\u53d8\u6362.<br \/>\n$\\\\$\u53cc\u66f2\u6811\u4f7f\u7528\u7684\u53cc\u66f2\u53d8\u6362\u4e3a\u5173\u4e8e\u590d\u6570$z$\u7684\u590d\u53d8\u51fd\u6570, \u5b83\u5177\u6709\u5982\u4e0b\u5f62\u5f0f:$$z_{(P, \\theta)} = \\frac{\\theta z + P}{1 + \\overline{P}z},$$\u5176\u4e2d, $P$\u4e0e$\\theta$\u5747\u4e3a\u590d\u6570, $|P| < 1$\u4e14$|\\theta| < 1$, $\\overline{P}$\u8868\u793a$P$\u7684\u5171\u8f6d\u590d\u6570. \u4e2a\u4eba\u8ba4\u4e3a\u6b64\u5904\u5206\u6bcd\u5904\u76841\u5e94\u4e3a1\u4e2a\u590d\u6570, \u56e0\u4e3a$\\overline{P}z$\u5e76\u4e0d\u4e00\u5b9a\u662f\u4e00\u4e2a\u5b9e\u6570. \u8be5\u53d8\u6362\u7684\u51e0\u4f55\u610f\u4e49\u662f\u5c06\u590d\u6570$z$\u7ed5\u539f\u70b9\u65cb\u8f6c\u590d\u6570$\\theta$\u5bf9\u5e94\u7684\u65cb\u8f6c\u89d2\u5ea6, \u7136\u540e\u5c06\u539f\u70b9\u79fb\u81f3$P$\u70b9. \u4f46\u5b58\u5728\u7591\u95ee\u7684\u4e00\u70b9\u662f, \u8be5\u53d8\u6362\u5e76\u4e0d\u6ee1\u8db3\u4e0a\u9762\u63d0\u5230\u7684\u53cc\u66f2\u7b49\u8ddd\u53d8\u6362\u5f62\u5f0f.\n$\\\\$\u4e24\u6b21\u53cc\u66f2\u53d8\u6362$z_{(P_1, \\theta_1)}, z_{(P_2, \\theta_2)}$\u7684\u590d\u5408\u53cc\u66f2\u53d8\u6362$z_{(P, \\theta)}$\u4e2d\u7684$P, \\theta$\u8ba1\u7b97\u5982\u4e0b:$$P = \\frac{\\theta_2 P_1 + P_2}{\\theta_2 P_1 \\overline{P_2} + 1}, \\\\ \\theta = \\frac{\\theta_1 \\theta_2 + \\theta_1 \\overline{P_1} P_2}{\\theta_2 P_1 \\overline{P_2} + 1}.$$\u7531\u4e8e\u820d\u5165\u8bef\u5dee, \u65b0\u5f97\u5230\u7684$\\theta$\u7684\u6a21\u4e0d\u4f1a\u4e25\u683c\u7b49\u4e8e1, \u7279\u522b\u662f\u5728\u5bf9\u8fb9\u754c\u9644\u8fd1\u7684\u70b9\u8fdb\u884c\u53d8\u6362\u65f6\u8fd9\u4e2a\u95ee\u9898\u4f1a\u88ab\u653e\u5927, \u56e0\u6b64\u603b\u662f\u9700\u8981\u5bf9\u65b0\u5f97\u5230\u7684$\\theta$\u8fdb\u884c\u91cd\u65b0\u5f52\u4e00\u5316.\n$\\\\$\u6b64\u5916, \u7531\u4e8e\u56fe\u5f62\u786c\u4ef6\u652f\u6301$3 \\times 3$\u7684\u77e9\u9635\u4e58\u6cd5, \u4f7f\u7528Klein\u6a21\u578b(\u4ea6\u79f0\u5c04\u5f71\u6a21\u578b) \u80fd\u591f\u83b7\u5f97\u66f4\u9ad8\u7684\u8ba1\u7b97\u6027\u80fd, \u56e0\u4e3a\u521a\u6027\u53d8\u6362\u53ef\u4ee5\u5229\u7528\u9f50\u6b21\u5750\u6807\u4e0a\u7684\u7ebf\u6027\u8fd0\u7b97\u8fdb\u884c\u8868\u793a. \u63a5\u4e0b\u6765\u53ef\u901a\u8fc7\u4e24\u79cd\u65b9\u5f0f\u5c06Klein\u6a21\u578b\u4e0a\u7684\u70b9\u91cd\u65b0\u6620\u5c04\u56dePoincare\u5706\u76d8\u6a21\u578b\u4e0a.\n$\\\\$a) \u901a\u8fc7$r_p = r_k \/ (1 + \\sqrt{1 - r_k^2})$\u91cd\u65b0\u8ba1\u7b97\u8ddd\u79bb\u539f\u70b9\u7684\u8ddd\u79bb.\n$\\\\$b) \u8bbe$s$\u4e3aKelin\u6a21\u578b\u4e0a\u6a21\u5c0f\u4e8e1\u7684\u4e00\u70b9, \u5219Poincare\u5706\u76d8\u6a21\u578b\u4e0a\u7684\u5bf9\u5e94\u70b9\u4e3a$$u = \\frac{s}{1 + \\sqrt{1 - s \\cdot s}} = \\frac{(1 - \\sqrt{1 - s \\cdot s})s}{s \\cdot s}.$$\n<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u5468\u4e94\u53bb\u533b\u9662\u505a\u4e86\u4e00\u6b21\u4f53\u68c0\u590d\u67e5, \u63a5\u4e0b\u6765\u5c31\u8981\u5f00\u542f\u4e00\u4e2a\u6708\u7684\u55d1\u836f\u5386\u7a0b\u4e86. \u54ce, \u5e0c\u671b\u771f\u7684\u53ea\u55d1\u4e00\u4e2a\u6708\u53ed, \u8eab\u4f53\u662f\u9769\u547d\u7684\u672c &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2022\/06\/12\/hyperbolic_tree\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u53cc\u66f2\u6811<\/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":[15,5],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1836"}],"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=1836"}],"version-history":[{"count":30,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1836\/revisions"}],"predecessor-version":[{"id":3627,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/1836\/revisions\/3627"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=1836"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=1836"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=1836"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}