{"id":912,"date":"2021-07-01T11:31:44","date_gmt":"2021-07-01T03:31:44","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=912"},"modified":"2025-02-26T11:19:32","modified_gmt":"2025-02-26T03:19:32","slug":"real_projective_plane_and_homogeneous_coordinates1","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2021\/07\/01\/real_projective_plane_and_homogeneous_coordinates1\/","title":{"rendered":"\u5b9e\u5c04\u5f71\u5e73\u9762\u53ca\u56fe\u5f62\u5b66\u4e2d\u7684\u9f50\u6b21\u5750\u6807(\u4e00)"},"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>\u6700\u8fd1\u4e24\u5929\u4e0b\u73ed\u65f6\u95f4\u90fd\u5728\u665a\u4e0a\u5341\u4e8c\u70b9\u4ee5\u540e\u4e86, \u5b9e\u5728\u6709\u70b9\u5403\u4e0d\u6d88, \u5927\u6982\u8fd9\u5c31\u662f\u6e38\u620f\u884c\u4e1a\u7684\u5e38\u6001\u53ed\u2026\u2026 \u4e0a\u5348\u8bf7\u4e86\u4e2a\u5047\u4f11\u606f\u4e00\u4e0b, \u91cd\u65b0\u5b66\u4e60\u4e86\u4e00\u4e0b\u5c04\u5f71\u5e73\u9762\u53ca\u9f50\u6b21\u5750\u6807, \u6709\u4e86\u66f4\u6df1\u7684\u7406\u89e3, \u7279\u6b64\u8bb0\u5f55\u4e00\u4e0b~<br \/>\n$\\\\$ \u76f4\u89c2\u4e0a\u6765\u7406\u89e3, \u5c04\u5f71\u5e73\u9762\u5176\u5b9e\u662f\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u4e0a\u76f4\u7ebf\u4e0e\u5e73\u9762\u7684\u9ad8\u5ea6\u62bd\u8c61, \u56e0\u4e3a\u5c04\u5f71\u5e73\u9762\u4e0d\u80fd\u65e0\u81ea\u4ea4\u5730\u5d4c\u5165\u4e09\u7ef4\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4, \u56e0\u6b64\u5728\u4e09\u7ef4\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4\u4e2d, \u6211\u4eec\u5e76\u4e0d\u80fd\u5b8c\u7f8e\u5730\u8868\u73b0\u5c04\u5f71\u5e73\u9762, \u4e5f\u5c31\u662f\u5f88\u96be\u53bb\u60f3\u8c61\u5b83\u7a76\u7adf\u957f\u5565\u6837. \u8fd9\u91cc\u7684\u201c\u5b8c\u7f8e\u201d \u662f\u6307\u65e2\u4e0d\u80fd\u628a\u76f4\u7ebf\u201c\u63b0\u5f2f\u201d, \u4e5f\u4e0d\u80fd\u5f15\u5165\u201c\u65e0\u7a77\u8fdc\u70b9\u201d \u7b49\u4e0d\u76f4\u89c2\u7684\u6982\u5ff5, \u4f46\u4e09\u7ef4\u7a7a\u95f4\u4e0b\u7684\u5c04\u5f71\u5e73\u9762(\u7a7a\u95f4) \u53ef\u4ee5\u5d4c\u5165\u5230\u56db\u7ef4\u6b27\u51e0\u91cc\u5f97\u7a7a\u95f4, \u8fd9\u4e0e\u514b\u83b1\u56e0\u74f6\u7c7b\u4f3c.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/102022619\">\u5c04\u5f71\u5e73\u9762\u7684\u76f4\u89c2\u7406\u89e3<\/a><br \/>\n2. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/62247239\">\u8ba1\u7b97\u673a\u89c6\u89c9\u7b14\u8bb01\uff1a\u767d\u8bdd\u5c04\u5f71\u51e0\u4f55<\/a><br \/>\n3. <a href=\"https:\/\/www.jianshu.com\/p\/7e701d7bfd79\">\u5199\u7ed9\u5927\u5bb6\u770b\u7684\u201c\u900f\u89c6\u9664\u6cd5\u201d \u2014\u2014 \u9f50\u6b21\u5750\u6807\u548c\u6295\u5f71<\/a><br \/>\n4. <a href=\"https:\/\/www.bilibili.com\/video\/BV1X7411F744?p=4\">GAMES101-\u73b0\u4ee3\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u5165\u95e8-\u95eb\u4ee4\u742a P4<\/a><\/p>\n<p><strong>1. \u5c04\u5f71\u7a7a\u95f4<\/strong><\/p>\n<p>\u4ece\u7b49\u4ef7\u7c7b\u7684\u89d2\u5ea6\u6765\u770b, \u82e5\u4e24\u4e2a\u70b9\u5747\u5904\u4e8e\u8fc7\u539f\u70b9\u7684\u540c\u4e00\u76f4\u7ebf\u4e0a, \u5219\u5728\u5c04\u5f71\u7a7a\u95f4\u4e2d\u662f\u7b49\u4ef7\u7684, \u5747\u8868\u793a\u540c\u4e00\u4e2a\u70b9; \u82e5\u4e24\u4e2a\u5e73\u9762\u5e73\u884c, \u5219\u5728\u5c04\u5f71\u7a7a\u95f4\u4e2d\u4e5f\u662f\u7b49\u4ef7\u7684, \u5747\u8868\u793a\u540c\u4e00\u6761\u76f4\u7ebf. \u5c04\u5f71\u7a7a\u95f4\u7684\u4e00\u4e2a\u91cd\u8981\u6027\u8d28\u662f\u4efb\u4f55\u4e24\u6761\u76f4\u7ebf\u76f8\u4ea4\u4e8e\u4e00\u70b9(\u56e0\u4e3a\u5f15\u5165\u4e86\u65e0\u7a77\u8fdc\u70b9).<br \/>\n$\\\\$ \u4e3a\u4e86\u76f4\u89c2\u7406\u89e3, \u901a\u5e38\u4f1a\u4f7f\u7528\u7403\u9762\u6765\u53ef\u89c6\u5316\u5c04\u5f71\u7a7a\u95f4, \u5982\u8003\u8651\u7403\u5fc3\u4f4d\u4e8e\u539f\u70b9\u7684\u5355\u4f4d\u7403, \u5219\u6240\u6709\u8fc7\u539f\u70b9\u7684\u76f4\u7ebf\u5747\u548c\u8fd9\u4e00\u7403\u9762\u6709\u4e14\u67092\u4e2a\u4ea4\u70b9, \u5254\u9664\u5bf9\u5f84\u70b9\u540e\u4fbf\u53ef\u4ee5\u5f97\u5230\u5c04\u5f71\u7a7a\u95f4\u4e00\u4e00\u5bf9\u5e94\u7684\u6295\u5f71\u70b9. \u8fd9\u6837\u6211\u4eec\u4fbf\u53ef\u4ee5\u5229\u7528\u8be5\u6295\u5f71\u70b9\u5728\u7403\u9762\u4e0a\u7684\u6cd5\u5411\u53bb\u5b9a\u4e49\u8be5\u70b9\u7684\u9f50\u6b21\u5750\u6807. \u8fd9\u5728\u7406\u8bba\u4e0a\u4e5f\u662f\u6709\u4fdd\u8bc1\u7684, \u56e0\u4e3a\u5c04\u5f71\u7a7a\u95f4\u540c\u80da\u4e8e\u7403\u9762\u7c98\u5408\u5bf9\u5f84\u70b9.<br \/>\n$\\\\$ \u56e0\u4e3a\u76f4\u7ebf\u7684\u5b9a\u4e49\u662f\u6700\u77ed\u8def\u5f84(\u7684\u5ef6\u7533), \u800c\u5728\u7403\u9762\u4e0a\u6700\u77ed\u8def\u5f84\u662f\u4ee5\u5706\u5f27\u7684\u5f62\u5f0f\u5448\u73b0\u7684, \u56e0\u6b64\u5c04\u5f71\u7a7a\u95f4\u4e2d\u7684\u76f4\u7ebf\u5728\u7403\u9762\u6295\u5f71\u662f\u4e00\u6bb5\u5706\u5f27. \u5b9e\u9645\u4e0a, \u4efb\u610f\u4e00\u6bb5\u5706\u5f27\u662f\u534a\u7403\u9762\u548c\u5e73\u9762 $Ax + By + Cz = 0$\u7684\u4ea4\u96c6. \u6211\u4eec\u53ef\u4ee5\u5229\u7528\u8be5\u5e73\u9762\u7684\u6cd5\u5411$(A, B, C)$\u6765\u5b9a\u4e49\u76f4\u7ebf\u7684\u9f50\u6b21\u5750\u6807. \u7279\u522b\u5730, \u5f53$A = B = C = 0$\u65f6\u5b9a\u4e49\u7684\u76f4\u7ebf\u662f\u4e00\u6761\u65e0\u7a77\u8fdc\u7684\u76f4\u7ebf.<\/p>\n<p><strong>2. \u9f50\u6b21\u5750\u6807<\/strong><\/p>\n<p>\u5c04\u5f71\u7a7a\u95f4\u4e2d\u7684\u5750\u6807\u88ab\u79f0\u4e3a\u9f50\u6b21\u5750\u6807, \u5229\u7528\u9f50\u6b21\u5750\u6807, \u6211\u4eec\u53ef\u4ee5\u5c06\u70b9\u8fdb\u884c\u5206\u7c7b, \u5f53\u628a\u5c04\u5f71\u7a7a\u95f4\u5d4c\u5165\u5230\u56db\u7ef4\u7a7a\u95f4\u4e2d\u4ee5\u540e, \u6bcf\u4e2a\u70b9\u7684\u9f50\u6b21\u5750\u6807\u90fd\u53ef\u4ee5\u7528\u4e00\u4e2a\u56db\u7ef4\u5411\u91cf$(x, $$ y, z, w)$\u6765\u8868\u793a, \u5f53\u5c04\u5f71\u7a7a\u95f4\u4e2d\u4e24\u4e2a\u70b9\u7684$w$\u5206\u91cf\u76f8\u540c\u65f6, \u5219\u610f\u5473\u7740\u5b83\u4eec\u5728\u5c04\u5f71\u7a7a\u95f4\u7684\u540c\u4e00\u4e2a\u5e73\u9762\u4e0a. \u5728\u6570\u5b66\u4e0a, \u6ca1\u6709\u6240\u8c13\u7684\u201c\u4e0d\u6b63\u786e\u201d \u7684\u9f50\u6b21\u5750\u6807, \u4f7f\u7528\u9f50\u6b21\u5750\u6807\u65f6\u8ba9$w=1$\u4ec5\u4ec5\u662f\u7528\u4e8e\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u4e2d\u7684\u6295\u5f71\u53d8\u6362.<br \/>\n$\\\\$ \u63a5\u4e0b\u6765\u518d\u6d45\u8c08\u4e00\u4e0b\u56fe\u5f62\u5b66\u4e2d\u7684\u9f50\u6b21\u5750\u6807, \u8fd9\u4e3b\u8981\u7528\u4e8e\u9876\u70b9Shader\u4e0a\u7684\u6295\u5f71\u53d8\u6362. \u5f53\u5c06\u4e00\u4e2a\u70b9$(x,y,z,1)$\u8fdb\u884c\u6295\u5f71\u53d8\u6362\u4ee5\u540e\u5176$w$\u5206\u91cf\u4e00\u822c\u4f1a\u53d8\u5316, \u5982\u5728\u8fd1\u5e73\u9762$z = $$ n$\u4e0a\u7684\u70b9\u7ecf\u6295\u5f71\u53d8\u6362\u4ee5\u540e\u5176$w$\u5206\u91cf\u4f1a\u53d8\u4e3a$n$. \u5728\u9876\u70b9Shader\u6267\u884c\u5b8c\u6bd5\u4ee5\u540e\u4f1a\u8fdb\u884c\u900f\u89c6\u53d8\u6362, \u5373\u5c06\u6bcf\u4e00\u4e2a\u70b9\u7684\u9f50\u6b21\u5750\u6807$(x, y, z, w)$\u7684\u6bcf\u4e00\u4e2a\u5206\u91cf\u90fd\u9664\u4ee5$w$($w$\u4e00\u822c\u4e0d\u4e3a0), \u6b64\u65f6\u70b9\u7684\u9f50\u6b21\u5750\u6807\u53d8\u4e3a$(x\/w, y\/w, z\/w, 1)$, \u800c\u8fd9\u4f9d\u65e7\u5728\u5c04\u5f71\u7a7a\u95f4\u4e2d\u8868\u793a\u540c\u4e00\u4e2a\u70b9, \u6216\u8bb8\u8fd9\u4e5f\u662f\u56fe\u5f62\u5b66\u4e0a\u8fdb\u884c\u900f\u89c6\u53d8\u6362\u7684\u7406\u8bba\u4f9d\u636e\u53ed, \u80fd\u6a21\u62df\u4eba\u773c\u770b\u5230\u7684\u4e16\u754c. \u6700\u540e\u53ea\u53d6$(x\/w, y\/w, z\/w, 1)$\u7684\u524d\u4e24\u4e2a\u5206\u91cf\u6784\u6210\u4e00\u4e2a\u4e8c\u7ef4\u5411\u91cf, \u5373\u8f6c\u5316\u4e3a\u5c4f\u5e55\u4e0a\u7684\u4e00\u4e2a\u70b9(\u867d\u7136\u540e\u9762\u8fd8\u9700\u8981\u8fdb\u884c\u89c6\u53e3\u53d8\u6362, \u56e0\u6b64\u8fd8\u4f1a\u6709\u4e00\u5b9a\u7684\u62c9\u4f38).<\/p>\n<p>PS: \u73b0\u4ee3\u6e32\u67d3\u6d41\u6c34\u7ebf\u8fdb\u884c\u900f\u89c6\u6295\u5f71\u7684\u7b97\u6cd5\u662f\u5148\u628a\u8fdc\u5e73\u9762\u8fdb\u884c\u538b\u7f29, \u628a\u89c6\u9525\u4f53\u538b\u7f29\u6210\u4e00\u4e2a\u957f\u65b9\u4f53, \u7136\u540e\u518d\u901a\u8fc7\u6b63\u4ea4\u6295\u5f71\u77e9\u9635\u53d8\u6362\u5230\u4e2d\u5fc3\u4f4d\u4e8e\u539f\u70b9\u7684\u8fb9\u957f\u4e3a2\u7684\u7acb\u65b9\u4f53$[-1, 1] \\times [-1, 1] \\times [-1, 1]$\u4e2d, \u5982\u4e0b\u56fe\u6240\u793a. \u76ee\u524d\u6211\u8fd8\u4e0d\u5927\u660e\u767d\u8fd9\u4e2a\u7b97\u6cd5\u4e3a\u4f55\u8981\u5c06\u900f\u89c6\u6295\u5f71\u8f6c\u5316\u4e3a\u6b63\u4ea4\u6295\u5f71\u7684\u539f\u56e0(\u76f4\u63a5\u5229\u7528<strong>\u628a<\/strong>\u5728\u8fd1\u5e73\u9762\u4e0a\u7684\u622a\u5f71\u4e0d\u9999\u4e48\u2026\u2026), \u6709\u5f85\u540e\u7eed\u7814\u7a76.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-676a506689164320a503d9323537825e_720w.jpg\" alt=\"\" width=\"711\" height=\"408\" class=\"alignnone size-full wp-image-918\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-676a506689164320a503d9323537825e_720w.jpg 711w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-676a506689164320a503d9323537825e_720w-300x172.jpg 300w\" sizes=\"(max-width: 711px) 100vw, 711px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u6700\u8fd1\u4e24\u5929\u4e0b\u73ed\u65f6\u95f4\u90fd\u5728\u665a\u4e0a\u5341\u4e8c\u70b9\u4ee5\u540e\u4e86, \u5b9e\u5728\u6709\u70b9\u5403\u4e0d\u6d88, \u5927\u6982\u8fd9\u5c31\u662f\u6e38\u620f\u884c\u4e1a\u7684\u5e38\u6001\u53ed\u2026\u2026 \u4e0a\u5348\u8bf7\u4e86\u4e2a\u5047\u4f11\u606f\u4e00\u4e0b &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/07\/01\/real_projective_plane_and_homogeneous_coordinates1\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u5b9e\u5c04\u5f71\u5e73\u9762\u53ca\u56fe\u5f62\u5b66\u4e2d\u7684\u9f50\u6b21\u5750\u6807(\u4e00)<\/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":[15,10],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/912"}],"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=912"}],"version-history":[{"count":14,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/912\/revisions"}],"predecessor-version":[{"id":3631,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/912\/revisions\/3631"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=912"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=912"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=912"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}