{"id":942,"date":"2021-07-17T20:46:04","date_gmt":"2021-07-17T12:46:04","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=942"},"modified":"2025-02-26T11:19:23","modified_gmt":"2025-02-26T03:19:23","slug":"homogeneous_space_clipping","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2021\/07\/17\/homogeneous_space_clipping\/","title":{"rendered":"\u9f50\u6b21\u7a7a\u95f4\u88c1\u526a"},"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>\u8fd9\u5468\u8fce\u6765\u4e86\u4e45\u8fdd\u7684\u5468\u672b, \u80fd\u5728\u5bb6\u5b85\u4e24\u5929\u771f\u7684\u662f\u5e78\u798f\u54c8\u54c8! \u8fd9\u5468\u51c6\u5907\u7ed3\u675f\u4e86\u9f50\u6b21\u7a7a\u95f4\u90e8\u5206\u7684\u5b66\u4e60~ \u5728\u4f20\u7edf\u7684\u6e32\u67d3\u7ba1\u7ebf\u4e2d, \u9f50\u6b21\u7a7a\u95f4\u7684\u88c1\u526a\u662f\u53d1\u751f\u5728\u9876\u70b9Shader\u4ee5\u540e, \u900f\u89c6\u9664\u6cd5\u4e4b\u524d\u7684, \u76ee\u7684\u5c31\u662f\u5254\u9664\u89c6\u9525\u4f53\u4ee5\u5916\u7684\u7269\u4f53, \u51cf\u5c11\u4e0d\u5fc5\u8981\u7684\u6e32\u67d3.<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/162190576\">\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u8865\u51452\uff1a\u9f50\u6b21\u7a7a\u95f4\u88c1\u526a(Homogeneous Space Clipping)<\/a><br \/>\n2. <a href=\"https:\/\/blog.csdn.net\/softwarekid\/article\/details\/45055253\">\u9f50\u6b21\u7a7a\u95f4\u7684\u88c1\u526a<\/a><br \/>\n3. <a href=\"https:\/\/zhuanlan.zhihu.com\/p\/102758967\">\u4e00\u7bc7\u6587\u7ae0\u5f7b\u5e95\u5f04\u61c2\u9f50\u6b21\u88c1\u526a<\/a><\/p>\n<blockquote><p>\u56fe\u7247\u57fa\u672c\u6765\u81ea\u4e8e\u77e5\u4e4e\u6587\u7ae0\u4e0eCSDN\u535a\u5ba2<\/p><\/blockquote>\n<p><strong>1. \u4e09\u89d2\u7f51\u683c\u4e0e\u89c6\u9525\u4f53\u7684\u88c1\u526a<\/strong><\/p>\n<p>\u663e\u7136, \u5728\u8fdb\u884c\u4e09\u89d2\u9762\u7684\u88c1\u526a\u65f6\u9700\u8981\u5224\u65ad\u70b9\u548c\u88c1\u526a\u5e73\u9762\u7684\u5173\u7cfb, \u5982\u4e0b\u56fe\u6240\u793a:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-1fb2c7db5d90de0d600296c6826aeab4_r.png\" alt=\"\" width=\"928\" height=\"602\" class=\"alignnone size-full wp-image-944\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-1fb2c7db5d90de0d600296c6826aeab4_r.png 928w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-1fb2c7db5d90de0d600296c6826aeab4_r-300x195.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-1fb2c7db5d90de0d600296c6826aeab4_r-768x498.png 768w\" sizes=\"(max-width: 928px) 100vw, 928px\" \/><\/p>\n<p>\u5176\u4e2d, $Q_1, Q_2$\u4e3a\u4e09\u89d2\u9762\u4e0a\u7684\u4e24\u4e2a\u9876\u70b9, $P$\u662f\u88c1\u526a\u5e73\u9762\u4e0a\u7684\u70b9, $\\overrightarrow{n}$\u4e3a\u88c1\u526a\u5e73\u9762\u4e0a\u7684\u6cd5\u5411, \u4e0d\u59a8\u4ee4$$d_1 = (Q_1-P) \\cdot \\overrightarrow{n}, \\\\ d_2 = (Q_2-P) \\cdot \\overrightarrow{n}.$$\u663e\u7136, \u5f53$d_1$\u4e0e$d_2$\u5f02\u53f7\u65f6\u8868\u793a\u8fb9$Q_1 Q_2$\u4e0e\u88c1\u526a\u5e73\u9762\u76f8\u4ea4, \u4e0d\u59a8\u8bbe\u5176\u76f8\u4ea4\u70b9\u4e3a$I$. \u5219$$I = Q_1 + t(Q_2 &#8211; Q_1),$$\u5176\u4e2d, $t = |Q_1 I| \/ |Q_1 Q_2|$, \u8fd9\u53ef\u4ee5\u7531\u76f8\u4f3c\u4e09\u89d2\u5f62\u6027\u8d28\u53ef\u5f97: \u5728$I$\u70b9\u5904\u4f5c\u4e00\u6761\u88c1\u526a\u5e73\u9762\u4e0a\u7684\u5782\u7ebf, \u5219\u53ef\u5f97\u62e5\u6709\u4e00\u5bf9\u5bf9\u9876\u89d2\u7684\u76f8\u4f3c\u4e09\u89d2\u5f62.<br \/>\n$\\\\$ \u8fd9\u6837\u4e00\u6765, \u5bf9\u4e8e\u4efb\u4e00\u4e09\u89d2\u7f51\u683c\u800c\u8a00, \u90fd\u53ef\u4ee5\u8fdb\u884c\u9010\u8fb9\u904d\u5386, \u628a\u6bcf\u4e00\u6761\u8fb9\u7ecf\u8fc7\u88c1\u526a\u5e73\u9762\u88c1\u526a\u5f97\u5230\u7684\u90e8\u5206\u4fdd\u7559\u4e0b\u6765, \u5373\u53ef\u5f97\u5230\u6700\u7ec8\u7684\u7ecf\u88c1\u526a\u540e\u7684\u7f51\u683c. \u66f4\u8fdb\u4e00\u6b65\u5730, \u5f53\u8003\u8651\u4e09\u89d2\u7f51\u683c\u4e0e\u89c6\u9525\u4f53\u7684\u88c1\u526a\u65f6, \u53ea\u9700\u8981\u8003\u8651\u9010\u4e2a\u88c1\u526a\u5e73\u9762\u7684\u88c1\u526a\u5373\u53ef(\u51716\u4e2a\u88c1\u526a\u5e73\u9762). \u800c\u7531\u8fd9\u4e2a\u601d\u8def\u8bde\u751f\u7684\u7b97\u6cd5\u4e5f\u662f\u9f0e\u9f0e\u5927\u540d\u7684Sutherland-Hodgeman\u88c1\u526a\u7b97\u6cd5.<\/p>\n<p><strong>2. \u9f50\u6b21(\u5c04\u5f71) \u7a7a\u95f4\u4e0b\u7684\u88c1\u526a<\/strong><\/p>\n<p>\u4f20\u7edf\u7684\u6e32\u67d3\u7ba1\u7ebf\u4e2d, \u7ecf\u8fc7Vertex Shader\u4e0e\u900f\u89c6\u9664\u6cd5\u4ee5\u540e, \u4fbf\u4f1a\u5c06\u9876\u70b9\u53d8\u6362\u5230NDC\u4e2d, \u539f\u6765\u7684\u89c6\u9525\u4f53\u4e5f\u4f1a\u53d8\u6210\u6807\u51c6\u7acb\u65b9\u4f53$[-1, 1] \\times [-1 $$ , 1] \\times [-1, 1]$, \u6b64\u65f66\u4e2a\u88c1\u526a\u5e73\u9762\u4e3a$$x = 1, x = -1, y = 1, \\\\ y = -1, z = 1, z = -1.$$\u4f46\u9700\u8981\u6e05\u695a\u7684\u662f, \u6e32\u67d3\u7ba1\u7ebf\u5e76\u4e0d\u662f\u5728NDC\u4e2d\u8fdb\u884c\u88c1\u526a\u7684, \u5177\u4f53\u539f\u56e0\u4f1a\u5728\u4e0b\u6587\u9610\u8ff0. \u5b9e\u9645\u4e0a, \u6e32\u67d3\u7ba1\u7ebf\u662f\u5728\u9f50\u6b21\u7a7a\u95f4\u4e2d\u8fdb\u884c\u88c1\u526a\u7684. \u90a3\u4e48\u95ee\u9898\u5c31\u8f6c\u5316\u4e3a\u9700\u8981\u627e\u5230NDC\u4e2d\u76846\u4e2a\u88c1\u526a\u5e73\u9762\u5728\u9f50\u6b21\u7a7a\u95f4\u4e2d\u5bf9\u5e94\u7684\u88c1\u526a\u5e73\u9762. \u6839\u636e\u900f\u89c6\u9664\u6cd5\u6027\u8d28, \u6613\u77e5\u9f50\u6b21\u7a7a\u95f4\u4e2d\u76846\u4e2a\u88c1\u526a\u5e73\u9762\u4e3a$$w = x, w = -x, w = y, \\\\ w = -y, w = z, w = -z.$$\u63a5\u4e0b\u6765\u4ee5$w = x$\u88c1\u526a\u5e73\u9762\u4e3a\u4f8b, \u8ba8\u8bba\u4e00\u4e0b\u4e09\u89d2\u5f62\u8fb9\u4e0e$w=x$\u88c1\u526a\u5e73\u9762\u7684\u88c1\u526a\u8ba1\u7b97. \u5047\u8bbe\u4e09\u89d2\u5f62\u8fb9$Q_1 Q_2$\u4e0e$w = x$\u88c1\u526a\u5e73\u9762\u4ea4\u70b9\u4e3a$I$, \u63d2\u503c\u7cfb\u6570\u4e3a$t$, \u5219$I = Q_1 + $$ t(Q_2 &#8211; Q_1),$ \u5982\u4e0b\u56fe\u6240\u793a:<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-6c16e812e37acfd4365261bcb36a2b4d_r.jpg\" alt=\"\" width=\"923\" height=\"568\" class=\"alignnone size-full wp-image-954\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-6c16e812e37acfd4365261bcb36a2b4d_r.jpg 923w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-6c16e812e37acfd4365261bcb36a2b4d_r-300x185.jpg 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-6c16e812e37acfd4365261bcb36a2b4d_r-768x473.jpg 768w\" sizes=\"(max-width: 923px) 100vw, 923px\" \/><\/p>\n<p>\u53c8\u4ea4\u70b9$I$\u843d\u5728\u88c1\u526a\u5e73\u9762$w = x$\u4e0a, \u56e0\u6b64\u4ea4\u70b9$I$\u7684$x$\u5206\u91cf\u4e0e$w$\u5206\u91cf\u76f8\u7b49, \u5373$$x_1 + t(x_2 &#8211; x_1) = w_1 + t(w_2 &#8211; w_1),$$\u89e3\u5f97$t = (x_1 &#8211; w_1) \/ ((w_2 &#8211; w_1) &#8211; (x_2 &#8211; x_1)).$ \u7c7b\u4f3c\u53ef\u5f97\u5176\u5b83\u88c1\u526a\u5e73\u9762\u7684\u4ea4\u70b9\u8ba1\u7b97\u7ed3\u679c.<\/p>\n<p>PS: \u9664\u4e86\u8003\u8651\u4e0a\u8ff06\u4e2a\u88c1\u526a\u5e73\u9762\u4e4b\u5916, \u901a\u5e38\u8fd8\u4f1a\u589e\u52a0\u4e00\u4e2a\u5173\u4e8e$w = $$ 10^{-5}$\u88c1\u526a\u5e73\u9762\u7684\u88c1\u526a, \u9632\u6b62\u5728\u8fdb\u884c\u900f\u89c6\u9664\u6cd5\u65f6\u51fa\u73b0\u96640\u9519\u8bef.<\/p>\n<p><strong>3. \u88c1\u526a\u540e\u7684\u9876\u70b9\u7ec4\u5408<\/strong><\/p>\n<p>\u4e0a\u8ff0\u88c1\u526a\u8fc7\u7a0b\u540e, \u4e00\u822c\u5f97\u5230\u7684\u53ea\u662f\u88c1\u526a\u540e\u7684\u9876\u70b9, \u5df2\u7ecf\u4e22\u5931\u4e86\u4e09\u89d2\u5f62\u7684\u4fe1\u606f. \u56e0\u6b64\u9700\u8981\u5c06\u9876\u70b9\u91cd\u65b0\u7ec4\u5408\u6210\u82e5\u5e72\u4e2a\u4e09\u89d2\u5f62, \u5176\u4e2d\u7684\u7ec4\u5408\u987a\u5e8f\u662f\u5341\u5206\u91cd\u8981\u7684, \u4e00\u822c\u4e0e\u539f\u4e09\u89d2\u5f62\u7684\u9876\u70b9\u987a\u5e8f\u4fdd\u6301\u4e00\u81f4(\u987a\u65f6\u9488\u65b9\u5411\u6216\u9006\u65f6\u9488\u65b9\u5411), \u8fd9\u4e5f\u662f\u4e3a\u4e86\u9632\u6b62\u5728\u8fdb\u884c\u6b63\u9762\u5254\u9664\u6216\u8005\u80cc\u9762\u5254\u9664\u7684\u65f6\u5019\u51fa\u73b0\u5947\u602a\u7684\u7ed3\u679c. \u5982\u4e0b\u56fe\u6240\u793a\u662f\u4e00\u79cd\u975e\u5e38\u7b80\u5355\u7684\u7ec4\u5408\u4e09\u89d2\u5f62\u7684\u65b9\u6cd5, \u8fd9\u4e5f\u662f\u6211\u4e4b\u524d\u5728\u8fdb\u884c\u70b9\u4e91\u91cd\u91c7\u6837\u7814\u7a76\u8ba1\u7b97\u5c40\u90e8\u6700\u4f73\u4f30\u8ba1\u7f51\u683c\u65f6\u91c7\u7528\u7684\u7f51\u683c\u5316\u65b9\u6cd5, \u4f46\u6211\u5e76\u4e0d\u786e\u5b9a\u73b0\u4ee3\u56fe\u5f62API\u7ec4\u5408\u4e09\u89d2\u5f62\u7684\u7b97\u6cd5, \u5176\u6548\u679c\u6709\u5f85\u8fdb\u4e00\u6b65\u9a8c\u8bc1.<\/p>\n<p style=\"text-align: center;\">\n<img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/\u56fe\u72471.png\" alt=\"\" width=\"256\" height=\"256\" class=\"alignnone size-full wp-image-956\" \/>\n<\/p>\n<p><strong>4. \u88c1\u526a\u7684\u91cd\u8981\u6027<\/strong><\/p>\n<p>\u9664\u4e86\u51cf\u5c11\u6e32\u67d3\u6d88\u8017\u4ee5\u5916, \u88c1\u526a\u8fd8\u6709\u4e00\u4e2a\u91cd\u8981\u610f\u4e49: \u9632\u6b62\u51fa\u73b0\u5947\u602a\u7684\u6e32\u67d3\u7ed3\u679c. \u5728\u89c2\u5bdf\u7a7a\u95f4\u4e2d, \u82e5\u4e00\u4e2a\u70b9\u7684\u5750\u6807\u662f$(x,y,z,1)$, \u901a\u8fc7\u6295\u5f71\u53d8\u6362\u4ee5\u540e\u5f97\u5230\u5176\u9f50\u6b21\u5750\u6807$(x&#8217; $$ , y&#8217;, z&#8217;, -z)$, \u518d\u7ecf\u900f\u89c6\u9664\u6cd5\u5f97\u5230$(x&#8221;, y&#8221;, $$ z&#8221;, w&#8221;) = (x&#8217;\/-z, y&#8217;\/-z, z&#8217; \/ $$ -z, 1)$. \u663e\u7136, \u5982\u679c\u5f53\u4e00\u4e2a\u70b9\u5728\u89c2\u5bdf\u70b9\u540e\u9762, \u5176\u89c2\u5bdf\u5750\u6807\u7684$z$\u5206\u91cf\u5927\u4e8e0(\u5047\u8bbe\u89c2\u5bdf\u70b9\u4f4d\u4e8e\u89c2\u5bdf\u7a7a\u95f4\u539f\u70b9, \u4e14\u9075\u5faa\u53f3\u624b\u539f\u5219, \u5373$z$\u8f74\u671d\u5411\u5c4f\u5e55\u5916), \u90a3\u4e48\u7ecf\u6295\u5f71\u53d8\u6362\u4ee5\u540e\u4ee5\u540e\u5f97\u5230\u7684\u9f50\u6b21\u5750\u6807\u7684$w$\u5206\u91cf\u5c06\u5c0f\u4e8e0, \u4ece\u800c\u7ecf\u900f\u89c6\u9664\u6cd5\u4ee5\u540e$(x&#8221;, y&#8221;, z&#8221;, $$ w&#8221;)$\u7684$x, y, z$\u5206\u91cf\u90fd\u5747\u53d8\u53f7, \u5bfc\u81f4\u90e8\u5206\u4e09\u89d2\u7f51\u683c\u53d1\u751f\u4e0a\u4e0b\u5de6\u53f3\u7ffb\u8f6c\u7684\u73b0\u8c61, \u5982\u4e0b\u56fe\u6240\u793a.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-fea83d5ed062c6bfe5dc8221e2d5d999_r.jpg\" alt=\"\" width=\"1688\" height=\"1065\" class=\"alignnone size-full wp-image-957\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-fea83d5ed062c6bfe5dc8221e2d5d999_r.jpg 1688w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-fea83d5ed062c6bfe5dc8221e2d5d999_r-300x189.jpg 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-fea83d5ed062c6bfe5dc8221e2d5d999_r-768x485.jpg 768w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/v2-fea83d5ed062c6bfe5dc8221e2d5d999_r-1536x969.jpg 1536w\" sizes=\"(max-width: 1688px) 100vw, 1688px\" \/><\/p>\n<p><strong>5. \u4e3a\u4ec0\u4e48\u4e0d\u5728\u6295\u5f71\u9664\u6cd5\u540e\u8fdb\u884c\u88c1\u526a<\/strong><\/p>\n<p>\u5148\u770b\u770b\u4e00\u822c\u7684\u6295\u5f71\u77e9\u9635\u5b9a\u4e49:<br \/>\n$$\\begin{bmatrix}<br \/>\ncot(\\theta\/2)\/r &#038; 0 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; cot(\\theta\/2) &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; (N + F) \/ (N &#8211; F) &#038; -2NF\/(N &#8211; F)\\\\<br \/>\n0 &#038; 0 &#038; -1 &#038; 0\\\\<br \/>\n\\end{bmatrix},$$\u5176\u4e2d, $\\theta$\u662f\u76f8\u673a\u7684FOV, $z = N$\u8868\u793a\u8fd1\u5e73\u9762, $z = F$\u8868\u793a\u8fdc\u5e73\u9762, $r$\u662f\u8fd1\u5e73\u9762$z = N$\u7684\u957f\u5bbd\u6bd4.<br \/>\n$\\\\$ \u5bf9\u4e8e\u89c2\u5bdf\u7a7a\u95f4\u4e2d\u7684\u70b9$(x, y, z, 1)$, \u7ecf\u6295\u5f71\u77e9\u9635\u7684\u4f5c\u7528\u540e\u5f97\u5230\u70b9\u7684\u9f50\u6b21\u5750\u6807$(x&#8217;, y&#8217; $$ , z&#8217;, -z)$, \u5176\u4e2d$x&#8217; = cot(\\theta\/2) \/ r \\cdot x$, \u663e\u7136$x&#8217;$\u4e0e$x$\u5448\u7ebf\u6027\u5173\u7cfb, \u540c\u7406\u53ef\u5f97$y&#8217;$\u4e0e$y$, $z&#8217;$\u4e0e$z$\u5747\u5448\u7ebf\u6027\u5173\u7cfb. \u4f46\u5728\u7ecf\u900f\u89c6\u9664\u6cd5\u4ee5\u540e, \u5f97\u5230\u5750\u6807$(x&#8221;, y&#8221;, z&#8221;, w&#8221;)$, \u6b64\u65f6$x&#8221; $$ = cot(\\theta\/2) \\cdot x \/ (r \\cdot -z)$\u662f\u5173\u4e8e$x$\u4e0e$z$\u7684\u51fd\u6570, \u4e0e$x$\u4e0d\u518d\u5448\u7ebf\u6027\u5173\u7cfb, \u540c\u7406\u53ef\u5f97$y&#8221;$\u4e0e$y$, $z&#8221;$\u4e0e$z$\u5747\u4e0d\u518d\u5448\u7ebf\u6027\u5173\u7cfb. \u82e5\u5728\u900f\u89c6\u9664\u6cd5\u4ee5\u540e\u8fdb\u884c\u5254\u9664, \u5219\u65e0\u6cd5\u518d\u4f7f\u7528\u7ebf\u6027\u63d2\u503c, \u5426\u5219\u70b9\u7684\u989c\u8272, \u7eb9\u7406\u7b49\u5c5e\u6027\u5728\u7ecf\u7ebf\u6027\u63d2\u503c\u4ee5\u540e\u90fd\u5c06\u4ea7\u751f\u5f02\u5e38.<\/p>\n<p><strong>6. \u4e3a\u4ec0\u4e48\u80fd\u5728\u9f50\u6b21\u7a7a\u95f4\u4e2d\u8fdb\u884c\u88c1\u526a<\/strong><\/p>\n<p>\u7531\u4e8e\u6295\u5f71\u77e9\u9635\u8868\u793a\u7684\u662f\u4e00\u7cfb\u5217\u4eff\u5c04\u53d8\u6362, \u800c\u4eff\u5c04\u53d8\u6362\u662f\u5e76\u4e0d\u6539\u53d8\u70b9\u7684\u989c\u8272, \u7eb9\u7406\u7b49\u5c5e\u6027\u7684, \u6545\u5728\u9f50\u6b21\u7a7a\u95f4\u4e0b\u88c1\u526a\u540e\u518d\u91c7\u7528\u7ebf\u6027\u63d2\u503c\u4e5f\u662f\u4e0d\u4f1a\u4ea7\u751f\u9876\u70b9\u5c5e\u6027\u7684\u5f02\u5e38\u7684. \u4f46\u8fd9\u91cc\u5c31\u6709\u4e00\u4e2a\u95ee\u9898: \u5728\u9f50\u6b21\u7a7a\u95f4\u4e2d\u88c1\u526a\u540e\u5f97\u5230\u7684\u70b9, \u7ecf\u8fc7\u900f\u89c6\u9664\u6cd5\u4ee5\u540e\u5728NDC\u4e2d\u6709\u6ca1\u6709\u53ef\u80fd\u88ab\u5254\u9664\u5462? \u7b54\u6848\u662f\u4e0d\u4f1a\u7684.<br \/>\n$\\\\$ \u4e3a\u4e86\u65b9\u4fbf\u5c55\u793a, \u4e0d\u59a8\u5c06$y, z$\u5206\u91cf\u56fa\u5b9a, \u4e0b\u56fe\u5c55\u793a\u4e86\u5206\u91cf$w$\u4e0e\u5206\u91cf$x$\u7684\u5173\u7cfb, \u56fe\u4e2d\u540c\u65f6\u5305\u542b\u4e86\u9f50\u6b21\u7a7a\u95f4\u4e0eNDC\u4e2d\u7684\u70b9, \u56fe\u4e2d\u6ca1\u6709\u753b\u51fa\u6765\u7684\u662f\u8f85\u52a9\u7ebf$w = x, w = -x$. \u6240\u8c13\u7684\u900f\u89c6\u9664\u6cd5\u5c31\u662f\u5c06\u9f50\u6b21\u7a7a\u95f4\u4e2d\u7684\u4e00\u70b9\u4e0e\u539f\u70b9\u76f8\u8fde\u540e\u5f97\u5230\u7684\u76f4\u7ebf\u4e0e$w = 1$\u6c42\u4ea4, \u4ece\u800c\u9f50\u6b21\u7a7a\u95f4\u4e2d\u7684\u70b9\u4e0e\u7ecf\u900f\u89c6\u9664\u6cd5\u540e\u843d\u5728$w = 1$\u4e0a\u7684\u70b9\u662f\u4e00\u4e00\u5bf9\u5e94\u7684. \u663e\u7136, \u56fe\u4e2d\u7684$P_0,P_1$\u5747\u4f4d\u4e8e$-x \\le w \\le x$\u533a\u57df\u5185, \u8bf4\u660e\u662f\u5728\u9f50\u6b21\u7a7a\u95f4\u4e2d\u7ecf\u88c1\u526a\u540e\u5f97\u5230\u7684\u70b9, \u90a3\u4e48\u5728\u7ecf\u8fc7\u900f\u89c6\u9664\u6cd5\u4ee5\u540e\u5f97\u5230\u7684\u70b9\u7684$x$\u5206\u91cf\u5fc5\u5b9a\u4e5f\u843d\u5728$[-1, 1]$\u5185; \u66f4\u8fdb\u4e00\u6b65\u5730, \u7ecf$P_0, P_1$\u7ebf\u6027\u63d2\u503c\u5f97\u5230\u7684\u70b9$P$, \u5fc5\u5b9a\u662f\u4f4d\u4e8e\u89c6\u9525\u4f53\u5185\u7684(\u89c6\u9525\u4f53\u662f\u4e00\u4e2a\u51f8\u591a\u9762\u4f53), \u5728\u7ecf\u8fc7\u900f\u89c6\u9664\u6cd5\u4ee5\u540e\u5f97\u5230\u7684\u70b9$P&#8217;$\u7684$x$\u5206\u91cf\u5fc5\u5b9a\u4e5f\u843d\u5728$[-1, 1]$\u5185, \u8bf4\u660e\u4e5f\u662f\u5728\u6807\u51c6\u7acb\u65b9\u4f53$[-1, 1] \\times [-1, 1] \\times $$ [-1, 1]$\u5185\u7684.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/20150418221359609.png\" alt=\"\" width=\"691\" height=\"564\" class=\"alignnone size-full wp-image-960\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/20150418221359609.png 691w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2021\/07\/20150418221359609-300x245.png 300w\" sizes=\"(max-width: 691px) 100vw, 691px\" \/><\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u8fd9\u5468\u8fce\u6765\u4e86\u4e45\u8fdd\u7684\u5468\u672b, \u80fd\u5728\u5bb6\u5b85\u4e24\u5929\u771f\u7684\u662f\u5e78\u798f\u54c8\u54c8! \u8fd9\u5468\u51c6\u5907\u7ed3\u675f\u4e86\u9f50\u6b21\u7a7a\u95f4\u90e8\u5206\u7684\u5b66\u4e60~ \u5728\u4f20\u7edf\u7684\u6e32\u67d3\u7ba1\u7ebf\u4e2d, &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2021\/07\/17\/homogeneous_space_clipping\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u9f50\u6b21\u7a7a\u95f4\u88c1\u526a<\/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":[10],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/942"}],"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=942"}],"version-history":[{"count":27,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/942\/revisions"}],"predecessor-version":[{"id":3630,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/942\/revisions\/3630"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=942"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=942"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=942"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}