{"id":3072,"date":"2023-03-19T22:10:25","date_gmt":"2023-03-19T14:10:25","guid":{"rendered":"https:\/\/www.caiqinyi.cn\/?p=3072"},"modified":"2025-02-26T11:01:25","modified_gmt":"2025-02-26T03:01:25","slug":"transformation_matrices_mark","status":"publish","type":"post","link":"https:\/\/www.caiqinyi.cn\/index.php\/2023\/03\/19\/transformation_matrices_mark\/","title":{"rendered":"\u53d8\u6362\u77e9\u9635\u6ce8\u8bb0"},"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>\u76ee\u524d\u5df2\u7ecf\u5728\u65b0\u516c\u53f8\u5de5\u4f5c\u4e86\u4e00\u5468\u591a\u4e86, \u524d\u9762\u4e24\u5929\u786e\u5b9e\u6709\u70b9\u5403\u4e0d\u6d88(\u5305\u62ec\u901a\u52e4\u65f6\u95f4\u4e0e\u5de5\u4f5c\u65f6\u95f4\u7b49\u56e0\u7d20), \u4f46\u73b0\u5728\u4e5f\u9010\u6e10\u9002\u5e94\u4e86\u76ee\u524d\u7684\u5de5\u4f5c\u8282\u594f. \u867d\u7136\u5e72\u7684\u6d3b\u57fa\u672c\u8fd8\u662f\u810f\u6d3b, \u4f46\u5bf9\u672a\u6765\u603b\u5f52\u8fd8\u662f\u6709\u76fc\u5934, \u5e0c\u671b\u672a\u6765\u8fd8\u662f\u80fd\u63a5\u51e0\u4f55\u5f62\u72b6\u5206\u6790\u76f8\u5173\u7684\u5de5\u4f5c\u5185\u5bb9\u53ed(\u4f8b\u5982PCG\u4e0ePVS)~ \u8fd9\u5468\u7684\u535a\u5ba2\u8fdb\u5ea6\u6709\u70b9\u6162, \u672c\u6765\u4ee5\u4e3a\u80fd\u5feb\u901f\u89e3\u51b3\u6389\u7ebf\u6027\u4ee3\u6570\u76f8\u5173\u7684\u5185\u5bb9, \u7ed3\u679c\u2026\u2026\u56e7( \u256f\u25a1\u2570 ) \u53d8\u6362\u77e9\u9635\u867d\u7136\u7528\u5f97\u6bd4\u8f83\u591a\u4e86, \u4f46\u8bfb\u5b8cShirley P, Ashikhmin M, Marschner S. Fundamentals of computer graphics[M]. AK Peters\/CRC Press, 2009.\u7684\u7b2c6\u7ae0, \u72b9\u5982\u918d\u9190\u704c\u9876, \u4eff\u4f5b\u53d1\u73b0\u4e86\u4e00\u4e2a\u65b0\u4e16\u754c~ \u56e0\u6b64, \u672c\u6587\u662f\u4e0d\u5f97\u4e0d\u5c06\u8fd9\u4e9b\u95ea\u5149\u70b9\u8bb0\u5f55\u4e0b\u6765\u6ef4!<\/p>\n<p><!--more--><\/p>\n<p><strong>\u53c2\u8003\u6750\u6599<\/strong><br \/>\n1. <a href=\"https:\/\/www.quora.com\/How-can-you-show-that-the-inverse-of-the-matrix-for-an-affine-transformation-one-that-has-all-zeros-in-the-bottom-row-except-for-a-one-in-the-lower-right-entry-also-has-the-same-form\">How can you show that the inverse of the matrix for an affine transformation (one that has all zeros in the bottom row except for a one in the lower right entry) also has the same form?<\/a><br \/>\n2. <a href=\"https:\/\/en.wikipedia.org\/wiki\/Affine_transformation\"> Affine transformation<\/a><\/p>\n<p><strong>1. 2\u7ef4\u7ebf\u6027\u53d8\u6362<\/strong><\/p>\n<p>\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a$2 \\times 2$\u7684\u77e9\u9635\u6765\u53d8\u6362\u4e00\u4e2a2\u7ef4\u5411\u91cf:$$\\begin{bmatrix}<br \/>\na_{11} &#038; a_{12}\\\\<br \/>\na_{21} &#038; a_{22}<br \/>\n\\end{bmatrix}\\begin{bmatrix}<br \/>\nx \\\\<br \/>\ny<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\na_{11}x + a_{12}y \\\\<br \/>\na_{21}x + a_{22}y<br \/>\n\\end{bmatrix}.$$\u8fd9\u79cd\u5c06\u4e00\u4e2a2\u7ef4\u5411\u91cf\u901a\u8fc7\u7b80\u5355\u7684\u77e9\u9635\u4e58\u6cd5\u5f97\u5230\u53e6\u5916\u4e00\u4e2a2\u7ef4\u5411\u91cf\u7684\u53d8\u6362\u4e3a\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362.<\/p>\n<p><strong>1.1 \u7f29\u653e<\/strong><\/p>\n<p>\u6700\u57fa\u672c\u7684\u53d8\u6362\u4e3a\u6cbf\u7740\u5750\u6807\u8f74\u7684\u7f29\u653e, \u8fd9\u4e2a\u53d8\u6362\u53ef\u4ee5\u6539\u53d8\u5411\u91cf\u7684\u957f\u5ea6\u4e0e\u65b9\u5411:$$scale(s_x, s_y) = \\begin{bmatrix}<br \/>\ns_x &#038; 0\\\\<br \/>\n0 &#038; s_y<br \/>\n\\end{bmatrix}.$$\u5f53\u4e0a\u8ff0\u77e9\u9635\u4f5c\u7528\u5728\u7b1b\u5361\u5c14\u5750\u6807$(x, y)$\u540e\u53ef\u5f97:$$\\begin{bmatrix}<br \/>\ns_x &#038; 0\\\\<br \/>\n0 &#038; s_y<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nx \\\\<br \/>\ny<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\ns_x x \\\\<br \/>\ns_y y<br \/>\n\\end{bmatrix}.$$\u56e0\u6b64, \u6211\u4eec\u53ef\u4ee5\u4ece\u4e00\u4e2a\u8f74\u5bf9\u9f50\u7684\u7f29\u653e\u77e9\u9635\u4e2d\u76f4\u63a5\u5f97\u52302\u4e2a\u7f29\u653e\u7684\u6bd4\u4f8b\u56e0\u5b50.<\/p>\n<p><strong>1.2 \u526a\u5207<\/strong><\/p>\n<p>\u4ece\u76f4\u89c2\u4e0a\u6765\u7406\u89e3, \u526a\u5207\u53d8\u6362\u662f\u4e00\u79cd\u628a\u4f5c\u7528\u5bf9\u8c61\u63a8\u5f80\u4e00\u8fb9\u7684\u53d8\u6362. \u6c34\u5e73\u65b9\u5411\u4e0e\u7ad6\u76f4\u65b9\u5411\u4e0a\u7684\u526a\u5207\u77e9\u9635\u5206\u522b\u4e3a$$shear-x(s) = \\begin{bmatrix}<br \/>\n1 &#038; s\\\\<br \/>\n0 &#038; 1<br \/>\n\\end{bmatrix}, shear-y(s) = \\begin{bmatrix}<br \/>\n1 &#038; 0\\\\<br \/>\ns &#038; 1<br \/>\n\\end{bmatrix}.$$\u53e6\u4e00\u79cd\u7406\u89e3\u526a\u5207\u7684\u89d2\u5ea6\u662f\u53ea\u8003\u8651\u7ad6\u76f4\u8f74(\u6216\u6c34\u5e73\u8f74) \u7684\u65cb\u8f6c. \u6c34\u5e73\u65b9\u5411\u4e0a\u7684\u526a\u5207\u53d8\u6362\u4e3a\u53d6\u7ad6\u76f4\u8f74\u5e76\u987a\u65f6\u9488\u503e\u659c\u89d2\u5ea6$\\phi$:$$\\begin{bmatrix}<br \/>\n1 &#038; tan \\phi\\\\<br \/>\n0 &#038; 1<br \/>\n\\end{bmatrix}.$$\u540c\u7406, \u7ad6\u76f4\u65b9\u5411\u4e0a\u7684\u526a\u5207\u53d8\u6362\u4e3a\u53d6\u6c34\u5e73\u8f74\u5e76\u9006\u65f6\u9488\u503e\u659c\u89d2\u5ea6$\\phi$:$$\\begin{bmatrix}<br \/>\n1 &#038; 0\\\\<br \/>\ntan \\phi &#038; 1<br \/>\n\\end{bmatrix}.$$<br \/>\n<strong>1.3 \u65cb\u8f6c<\/strong><\/p>\n<p>\u8bbe\u5411\u91cf$\\mathbf{a}$\u4e0e$x$\u8f74\u7684\u5939\u89d2\u4e3a$\\alpha$, \u5176\u957f\u5ea6\u4e3a$r = x_\\mathbf{a}^2 + y_\\mathbf{a}^2$, $\\mathbf{b}$\u4e3a\u5411\u91cf$\\mathbf{a}$\u9006\u65f6\u9488\u65cb\u8f6c\u89d2\u5ea6$\\phi$\u5f97\u5230\u7684\u5411\u91cf, \u5219$$x_\\mathbf{a} = rcos\\alpha, \\\\ y_\\mathbf{a} = rsin\\alpha.$$\u7531\u4e8e\u5411\u91cf$\\mathbf{b}$\u662f\u901a\u8fc7\u5c06\u5411\u91cf$\\mathbf{a}$\u9006\u65f6\u9488\u65cb\u8f6c\u89d2\u5ea6$\\phi$\u5f97\u5230\u7684, \u6545\u5176\u957f\u5ea6\u4e5f\u4e3a$r$, \u4e0e$x$\u8f74\u7684\u5939\u89d2\u4e3a$(\\alpha + \\phi)$. \u7531\u4e09\u89d2\u51fd\u6570\u7684\u79ef\u5316\u548c\u5dee\u516c\u5f0f\u53ef\u5f97:$$x_\\mathbf{b} = rcos(\\alpha + \\phi) = r cos\\alpha cos\\phi &#8211; r sin\\alpha sin\\phi, \\\\ y_\\mathbf{b} = rsin(\\alpha + \\phi) = r sin\\alpha cos\\phi + r cos\\alpha sin\\phi.$$\u4ee3\u5165$x_\\mathbf{a} = rcos\\alpha$\u4e0e$y_\\mathbf{a} = rsin\\alpha$\u53ef\u5f97$$x_\\mathbf{b} = x_\\mathbf{a} cos\\phi &#8211; y_\\mathbf{a} sin\\phi, \\\\ y_\\mathbf{b} = y_\\mathbf{a} cos\\phi + x_\\mathbf{a} sin\\phi.$$\u91c7\u7528\u77e9\u9635\u7684\u8bed\u8a00\u63cf\u8ff0\u4e0a\u8ff0\u53d8\u6362,$$rotate(\\phi) = \\begin{bmatrix}<br \/>\ncos\\phi &#038; -sin\\phi\\\\<br \/>\nsin\\phi &#038; cos\\phi<br \/>\n\\end{bmatrix}.$$<br \/>\n<strong>1.4 \u53cd\u5c04<\/strong><\/p>\n<p>\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u4f7f\u7528\u5e26\u6709\u4e00\u4e2a\u8d1f\u6bd4\u4f8b\u56e0\u5b50\u7684\u77e9\u9635, \u6765\u5f97\u5230\u5173\u4e8e\u7279\u5b9a\u5750\u6807\u8f74\u7684\u53cd\u5c04\u5411\u91cf:$$reflect-y = \\begin{bmatrix}<br \/>\n-1 &#038; 0\\\\<br \/>\n0 &#038; 1<br \/>\n\\end{bmatrix}, relfect-x = \\begin{bmatrix}<br \/>\n1 &#038; 0\\\\<br \/>\n0 &#038; -1<br \/>\n\\end{bmatrix}.$$<br \/>\n<strong>1.5 \u53d8\u6362\u7684\u7ec4\u5408<\/strong><\/p>\n<p>\u5728\u56fe\u5f62\u7a0b\u5e8f\u4e2d, \u5bf9\u4e00\u4e2a\u5bf9\u8c61\u5e94\u7528\u591a\u4e2a\u53d8\u6362\u662f\u5341\u5206\u5e38\u89c1\u7684. \u4f8b\u5982, \u6211\u4eec\u53ef\u80fd\u60f3\u5bf9\u4e00\u4e2a2\u7ef4\u5411\u91cf$\\mathbf{v}_1$\u5148\u5e94\u7528\u7f29\u653e\u77e9\u9635$\\mathbf{S}$, \u518d\u5e94\u7528\u65cb\u8f6c\u77e9\u9635$\\mathbf{R}$, \u8fd9\u53ef\u4ee5\u5206\u4e24\u6b65\u5b8c\u6210:$$\\mathbf{v}_2 = \\mathbf{S} \\mathbf{v}_1, \\mathbf{v}_3 = \\mathbf{R} \\mathbf{v}_2.$$\u53e6\u4e00\u79cd\u5199\u6cd5\u662f$$\\mathbf{v}_3 = \\mathbf{R} (\\mathbf{S} \\mathbf{v}_1).$$\u7531\u4e8e\u77e9\u9635\u4e58\u6cd5\u662f\u6ee1\u8db3\u7ed3\u5408\u5f8b\u7684, \u6545\u6211\u4eec\u4e5f\u53ef\u4ee5\u5199\u6210$$\\mathbf{v}_3 = (\\mathbf{R} \\mathbf{S})\\mathbf{v}_1.$$\u6362\u53e5\u8bdd\u8bf4, \u6211\u4eec\u53ef\u4ee5\u7528\u4e00\u4e2a\u76f8\u540c\u5927\u5c0f\u7684\u77e9\u9635\u6765\u8868\u793a\u7528\u4e24\u4e2a\u77e9\u9635\u4f9d\u6b21\u53d8\u6362\u4e00\u4e2a\u5411\u91cf\u7684\u6548\u679c, \u5373\u53ef\u4ee5\u901a\u8fc7\u5c06\u4e24\u4e2a\u77e9\u9635\u76f8\u4e58\u6765\u8ba1\u7b97:$$\\mathbf{M} = \\mathbf{R} \\mathbf{S}.$$\u9700\u8981\u6ce8\u610f\u7684\u662f, \u8fd9\u4e9b\u53d8\u6362\u662f\u4ece\u53f3\u8fb9\u5f00\u59cb\u5e94\u7528\u7684. \u5bf9\u4e8e\u77e9\u9635$\\mathbf{M} = \\mathbf{R} \\mathbf{S}$\u800c\u8a00, \u6211\u4eec\u9996\u5148\u5e94\u7528\u7f29\u653e\u53d8\u6362$\\mathbf{S}$, \u7136\u540e\u5e94\u7528\u65cb\u8f6c\u53d8\u6362$\\mathbf{R}$.<\/p>\n<p><strong>1.6 \u53d8\u6362\u7684\u5206\u89e3<\/strong><\/p>\n<p><strong>$\\cdot$ \u5b9e\u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u5206\u89e3<\/strong><\/p>\n<p>\u6211\u4eec\u4ece\u5b9e\u5bf9\u79f0\u77e9\u9635\u5f00\u59cb. \u4e00\u4e2a\u5b9e\u5bf9\u79f0\u77e9\u9635\u603b\u53ef\u4ee5\u8fdb\u884c\u7279\u5f81\u503c\u5206\u89e3$$\\mathbf{A} = \\mathbf{R} \\mathbf{S} \\mathbf{R}^T,$$\u5176\u4e2d, $\\mathbf{R}$\u4e3a\u6b63\u4ea4\u77e9\u9635, $\\mathbf{S}$\u4e3a\u5bf9\u89d2\u77e9\u9635; \u6211\u4eec\u79f0\u6b63\u4ea4\u77e9\u9635$\\mathbf{R}$\u7684\u5217\u5411\u91cf(\u7279\u5f81\u5411\u91cf) \u4e3a$\\mathbf{v}_1$\u4e0e$\\mathbf{v}_2$, \u79f0\u5bf9\u89d2\u77e9\u9635$\\mathbf{S}$\u7684\u5bf9\u89d2\u7ebf\u5143\u7d20(\u7279\u5f81\u503c) \u4e3a$\\lambda_1$\u4e0e$\\lambda_2$.<br \/>\n$\\\\$\u4ece\u51e0\u4f55\u4e0a\u6765\u770b, \u6211\u4eec\u53ef\u4ee5\u5c06\u6b63\u4ea4\u77e9\u9635$\\mathbf{R}$\u89c6\u4e3a\u65cb\u8f6c\u53d8\u6362, \u5c06\u5bf9\u89d2\u77e9\u9635$\\mathbf{S}$\u89c6\u4e3a\u7f29\u653e\u53d8\u6362, \u56e0\u6b64\u8fd9\u662f\u4e00\u4e2a\u591a\u6b65\u7684\u51e0\u4f55\u53d8\u6362:<br \/>\n$\\\\$ (1) \u5c06\u4e0e\u7279\u5f81\u5411\u91cf$\\mathbf{v}_1$\u5171\u7ebf\u7684\u5411\u91cf\u548c\u4e0e\u7279\u5f81\u5411\u91cf$\\mathbf{v}_2$\u5171\u7ebf\u7684\u5411\u91cf\u5206\u522b\u65cb\u8f6c\u5230$x$\u8f74\u4e0a\u4e0e$y$\u8f74\u4e0a(\u901a\u8fc7\u6b63\u4ea4\u77e9\u9635$\\mathbf{R}^T$\u8fdb\u884c\u53d8\u6362).<br \/>\n$\\\\$ (2) \u5206\u522b\u6cbf$x$\u8f74\u4e0e$y$\u8f74\u4ee5\u7f29\u653e\u56e0\u5b50$\\lambda_1$\u4e0e$\\lambda_2$\u8fdb\u884c\u7f29\u653e(\u901a\u8fc7\u5bf9\u89d2\u77e9\u9635$\\mathbf{S}$\u8fdb\u884c\u53d8\u6362).<br \/>\n$\\\\$ (3) \u5c06\u4e0e$x$\u8f74\u5171\u7ebf\u7684\u5411\u91cf\u548c\u4e0e$y$\u8f74\u5171\u7ebf\u7684\u5411\u91cf\u5206\u522b\u65cb\u8f6c\u81f3\u4e0e\u7279\u5f81\u5411\u91cf$\\mathbf{v}_1$\u5171\u7ebf\u548c\u4e0e\u7279\u5f81\u5411\u91cf$\\mathbf{v}_2$\u5171\u7ebf(\u901a\u8fc7\u6b63\u4ea4\u77e9\u9635$\\mathbf{R}$\u8fdb\u884c\u53d8\u6362).<br \/>\n$\\\\$ \u628a\u4e0a\u8ff03\u4e2a\u53d8\u6362\u653e\u5728\u4e00\u8d77\u770b, \u6211\u4eec\u53ef\u4ee5\u770b\u5230\u5b83\u4eec\u5728\u5b9e\u5bf9\u79f0\u77e9\u9635$\\mathbf{A}$\u7684\u7279\u5f81\u5411\u91cf\u5bf9\u5e94\u7684\u4e00\u5bf9\u5750\u6807\u8f74\u4e0a(\u4ea6\u662f\u76f8\u4e92\u5782\u76f4\u7684) \u5177\u6709\u4e0d\u5747\u5300\u5c3a\u5ea6\u7684\u6548\u679c. \u8fd9\u544a\u8bc9\u4e86\u6211\u4eec\u4ec0\u4e48\u662f\u5b9e\u5bf9\u79f0\u77e9\u9635: \u5b9e\u5bf9\u79f0\u77e9\u9635\u53ea\u662f\u5bf9\u5e94\u4e00\u4e2a\u7f29\u653e\u53d8\u6362\u2014\u2014\u5c3d\u7ba1\u53ef\u80fd\u662f\u975e\u5747\u5300\u7684\u4e0e\u975e\u8f74\u5bf9\u9f50\u7684.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2023\/03\/symmetric_matrix_transformation.png\" alt=\"\" width=\"833\" height=\"367\" class=\"aligncenter size-full wp-image-3095\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2023\/03\/symmetric_matrix_transformation.png 833w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2023\/03\/symmetric_matrix_transformation-300x132.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2023\/03\/symmetric_matrix_transformation-768x338.png 768w\" sizes=\"(max-width: 833px) 100vw, 833px\" \/><\/p>\n<p>\u6211\u4eec\u4e5f\u53ef\u4ee5\u53cd\u8f6c\u4e0a\u8ff0\u7684\u5bf9\u89d2\u5316\u8fc7\u7a0b; \u4ee5$(\\lambda_1, \\lambda_2)$\u4e3a\u7f29\u653e\u56e0\u5b50, \u4ece$x$\u8f74\u987a\u65f6\u9488\u65b9\u5411\u65cb\u8f6c\u89d2\u5ea6$\\phi$, \u5219\u6211\u4eec\u6709$$\\begin{bmatrix}<br \/>\ncos\\phi &#038; sin\\phi\\\\<br \/>\n-sin\\phi &#038; cos\\phi<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n\\lambda_1 &#038; 0\\\\<br \/>\n0 &#038; \\lambda_2<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\ncos\\phi &#038; -sin\\phi\\\\<br \/>\nsin\\phi &#038; cos\\phi<br \/>\n\\end{bmatrix} = \\\\<br \/>\n\\begin{bmatrix}<br \/>\n\\lambda_1 cos^2\\phi + \\lambda_2 sin^2\\phi &#038; (\\lambda_2 &#8211; \\lambda_1) cos\\phi sin\\phi\\\\<br \/>\n(\\lambda_2 &#8211; \\lambda_1) cos\\phi sin\\phi &#038; \\lambda_2 cos^2\\phi + \\lambda_1 sin^2\\phi<br \/>\n\\end{bmatrix}.$$\u503c\u5f97\u6ce8\u610f\u7684\u662f, \u8fd9\u662f\u4e00\u4e2a\u5bf9\u79f0\u77e9\u9635, \u56e0\u4e3a\u5176\u662f\u4ee5\u5b9e\u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u5206\u89e3\u7684\u5f62\u5f0f\u6784\u9020\u7684.<\/p>\n<p><strong>$\\cdot$ \u5947\u5f02\u503c\u5206\u89e3<\/strong><\/p>\n<p>\u4e00\u79cd\u975e\u5e38\u7c7b\u4f3c\u7684\u5206\u89e3\u4e5f\u53ef\u4ee5\u7528\u4e8e\u975e\u5bf9\u79f0\u77e9\u9635: \u5b83\u4fbf\u662f\u5947\u5f02\u503c\u5206\u89e3(SVD). \u5b83\u4e0e\u5b9e\u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u5206\u89e3\u7684\u4e0d\u540c\u4e4b\u5904\u5728\u4e8e\u5bf9\u89d2\u77e9\u9635\u4e24\u4fa7\u7684\u77e9\u9635\u4e0d\u518d\u8981\u6c42\u76f8\u540c:$$\\mathbf{A} = \\mathbf{U} \\mathbf{S} \\mathbf{V}^T.$$\u4ee3\u66ff\u5355\u6b21\u65cb\u8f6c\u77e9\u9635$\\mathbf{R}$\u7684\u4e24\u4e2a\u6b63\u4ea4\u77e9\u9635\u5206\u522b\u4e3a$\\mathbf{U}$\u4e0e$\\mathbf{V}$, \u5b83\u4eec\u7684\u5217\u5411\u91cf\u5206\u522b\u8bb0\u4e3a$\\mathbf{u}_i$(\u5de6\u5947\u5f02\u5411\u91cf) \u4e0e$\\mathbf{v}_i$(\u53f3\u5947\u5f02\u5411\u91cf). \u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b, \u5bf9\u89d2\u77e9\u9635$\\mathbf{S}$\u7684\u5bf9\u89d2\u7ebf\u5143\u7d20\u88ab\u79f0\u4e3a\u5947\u5f02\u503c\u800c\u975e\u7279\u5f81\u503c. \u5176\u51e0\u4f55\u89e3\u91ca\u4e0e\u5b9e\u5bf9\u79f0\u77e9\u9635\u7684\u7279\u5f81\u503c\u5206\u89e3\u975e\u5e38\u76f8\u4f3c:<br \/>\n$\\\\$ (1) \u5c06\u4e0e\u5947\u5f02\u5411\u91cf$\\mathbf{v}_1$\u5171\u7ebf\u7684\u5411\u91cf\u548c\u4e0e\u5947\u5f02\u5411\u91cf$\\mathbf{v}_2$\u5171\u7ebf\u7684\u5411\u91cf\u5206\u522b\u65cb\u8f6c\u5230$x$\u8f74\u4e0a\u4e0e$y$\u8f74\u4e0a(\u901a\u8fc7\u6b63\u4ea4\u77e9\u9635$\\mathbf{V}^T$\u8fdb\u884c\u53d8\u6362).<br \/>\n$\\\\$ (2) \u5206\u522b\u6cbf$x$\u8f74\u4e0e$y$\u8f74\u4ee5\u7f29\u653e\u56e0\u5b50$\\sigma_1$\u4e0e$\\sigma_2$\u8fdb\u884c\u7f29\u653e\u53d8\u6362(\u901a\u8fc7\u5bf9\u89d2\u77e9\u9635$\\mathbf{S}$\u8fdb\u884c\u53d8\u6362).<br \/>\n$\\\\$ (3) \u5c06\u4e0e$x$\u8f74\u5171\u7ebf\u7684\u5411\u91cf\u548c\u4e0e$y$\u8f74\u5171\u7ebf\u7684\u5411\u91cf\u5206\u522b\u65cb\u8f6c\u81f3\u4e0e\u5947\u5f02\u5411\u91cf$\\mathbf{u}_1$\u5171\u7ebf\u548c\u4e0e\u5947\u5f02\u5411\u91cf$\\mathbf{u}_2$\u5171\u7ebf(\u901a\u8fc7\u6b63\u4ea4\u77e9\u9635$\\mathbf{U}$\u8fdb\u884c\u53d8\u6362).<br \/>\n$\\\\$ \u4e3b\u8981\u7684\u533a\u522b\u4e3a\u4e00\u4e2a\u65cb\u8f6c\u77e9\u9635\u4e0e\u4e24\u4e2a\u4e0d\u540c\u7684\u6b63\u4ea4\u77e9\u9635\u4e4b\u95f4\u7684\u533a\u522b, \u8fd9\u79cd\u5dee\u5f02\u8bf1\u5bfc\u4e86SVD\u7684\u4e00\u4e2a\u53ef\u80fd\u4e0d\u8d77\u773c\u7684\u7279\u5f81. \u7531\u4e8eSVD\u7684\u4e24\u8fb9\u7684\u5947\u5f02\u5411\u91cf\u53ef\u80fd\u4e0d\u540c, \u6240\u4ee5\u4e0d\u9700\u8981\u8d1f\u5947\u5f02\u503c: \u6211\u4eec\u603b\u662f\u53ef\u4ee5\u53cd\u8f6c\u4e00\u4e2a\u5947\u5f02\u503c\u7684\u7b26\u53f7, \u518d\u53cd\u8f6c\u5176\u4e2d\u4e00\u4e2a\u5bf9\u5e94\u7684\u5947\u5f02\u5411\u91cf\u7684\u65b9\u5411, \u4ece\u800c\u518d\u6b21\u5f97\u5230\u76f8\u540c\u7684\u53d8\u6362. \u7531\u4e8e\u8fd9\u4e2a\u539f\u56e0, SVD\u5f97\u5230\u7684\u5bf9\u89d2\u77e9\u9635\u7684\u5bf9\u89d2\u7ebf\u5143\u7d20\u603b\u4e3a\u6b63\u6570, \u4f46\u6b63\u4ea4\u77e9\u9635$\\mathbf{U}$\u4e0e\u6b63\u4ea4\u77e9\u9635$\\mathbf{V}$\u65e0\u6cd5\u4fdd\u8bc1\u662f\u65cb\u8f6c\u77e9\u9635\u2014\u2014\u5b83\u4eec\u4ea6\u53ef\u80fd\u4e3a\u53cd\u5c04\u77e9\u9635. \u5728\u56fe\u5f62\u7a0b\u5e8f\u4e2d, \u8fd9\u5f88\u4e0d\u65b9\u4fbf, \u4f46\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u4e00\u4e2a\u5c0f\u6280\u5de7\u6765\u89c4\u907f\u8fd9\u4e2a\u95ee\u9898: \u901a\u8fc7\u68c0\u67e5\u884c\u5217\u5f0f, \u6211\u4eec\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u533a\u5206\u65cb\u8f6c\u77e9\u9635\u4e0e\u53cd\u5c04\u77e9\u9635, \u56e0\u4e3a\u65cb\u8f6c\u77e9\u9635\u7684\u884c\u5217\u5f0f\u4e3a+1, \u800c\u53cd\u5c04\u77e9\u9635\u7684\u884c\u5217\u5f0f\u4e3a\u22121. \u5982\u6b64\u4e00\u6765, \u82e5\u901a\u8fc7SVD\u5f97\u5230\u7684\u67d0\u4e2a\u6b63\u4ea4\u77e9\u9635(\u5982\u6b63\u4ea4\u77e9\u9635$\\mathbf{U}$) \u4e3a\u53cd\u5c04\u77e9\u9635, \u800c\u6211\u4eec\u5e0c\u671b\u5bf9\u89d2\u77e9\u9635$\\mathbf{S}$\u4e24\u4fa7\u7684\u6b63\u4ea4\u77e9\u9635\u5747\u4e3a\u65cb\u8f6c\u77e9\u9635, \u5219\u6211\u4eec\u53ef\u4ee5\u5c06\u5176\u4e2d\u4e00\u4e2a\u5947\u5f02\u503c\u7684\u7b26\u53f7\u8fdb\u884c\u53cd\u8f6c\u5f97\u5230\u65b0\u7684\u5bf9\u89d2\u77e9\u9635$\\mathbf{S}&#8217;$, \u540c\u65f6\u5c06\u6b63\u4ea4\u77e9\u9635$\\mathbf{U}$\u4e2d\u5bf9\u5e94\u7684\u5947\u5f02\u5411\u91cf\u7684\u65b9\u5411\u4e00\u5e76\u8fdb\u884c\u53cd\u8f6c\u5f97\u5230\u65b0\u7684\u6b63\u4ea4\u77e9\u9635$\\mathbf{U}&#8217;$, \u5219\u6b64\u65f6$\\mathbf{A} = \\mathbf{U}&#8217; \\mathbf{S}&#8217; \\mathbf{V}^T$, \u5bf9\u89d2\u77e9\u9635$\\mathbf{S}&#8217;$\u4e24\u4fa7\u7684\u6b63\u4ea4\u77e9\u9635\u5747\u4e3a\u65cb\u8f6c\u77e9\u9635.<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" src=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2023\/03\/arbitary_matrix_transformation.png\" alt=\"\" width=\"825\" height=\"368\" class=\"aligncenter size-full wp-image-3097\" srcset=\"https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2023\/03\/arbitary_matrix_transformation.png 825w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2023\/03\/arbitary_matrix_transformation-300x134.png 300w, https:\/\/www.caiqinyi.cn\/wp-content\/uploads\/2023\/03\/arbitary_matrix_transformation-768x343.png 768w\" sizes=\"(max-width: 825px) 100vw, 825px\" \/><\/p>\n<p><strong>$\\cdot$ \u65cb\u8f6c\u77e9\u9635\u7684Paeth\u5206\u89e3<\/strong><\/p>\n<p>\u975e\u96f6\u7684\u65cb\u8f6c\u77e9\u9635\u53ef\u4ee5\u5229\u7528\u526a\u5207\u53d8\u6362\u8fdb\u884c\u5206\u89e3(Paeth, 1990), \u5373$$\\begin{bmatrix}<br \/>\ncos\\phi &#038; sin\\phi\\\\<br \/>\n-sin\\phi &#038; cos\\phi<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\n1 &#038; \\frac{cos\\phi &#8211; 1}{sin\\phi}\\\\<br \/>\n0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n1 &#038; 0\\\\<br \/>\nsin\\phi &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n1 &#038; \\frac{cos\\phi &#8211; 1}{sin\\phi}\\\\<br \/>\n0 &#038; 1<br \/>\n\\end{bmatrix}.$$\u8fd9\u79cd\u7279\u6b8a\u7684\u53d8\u6362\u5bf9\u5149\u6805\u5316\u56fe\u50cf\u4e0a\u7684\u65cb\u8f6c\u53d8\u6362\u5f88\u6709\u7528, \u56e0\u4e3a\u526a\u5207\u53d8\u6362\u662f\u4e00\u79cd\u975e\u5e38\u6709\u6548\u7684\u56fe\u50cf\u5149\u6805\u64cd\u4f5c; \u5b83\u4f1a\u4ea7\u751f\u4e00\u4e9b\u952f\u9f7f, \u4f46\u4e0d\u4f1a\u7559\u4e0b\u6d1e. \u503c\u5f97\u6ce8\u610f\u7684\u662f, \u82e5\u6211\u4eec\u53d6\u5149\u6805\u5316\u56fe\u50cf\u4e0a\u7684\u4e00\u4e2a\u50cf\u7d20(\u5176\u5750\u6807\u4e3a$(i, j)$), \u5e76\u5bf9\u5176\u5e94\u7528\u6c34\u5e73\u65b9\u5411\u4e0a\u7684\u526a\u5207\u53d8\u6362, \u53ef\u5f97$$\\begin{bmatrix}<br \/>\n1 &#038; s\\\\<br \/>\n0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\ni \\\\<br \/>\nj<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\ni + sj \\\\<br \/>\nj<br \/>\n\\end{bmatrix}.$$\u82e5\u6211\u4eec\u5c06$sj$\u56db\u820d\u4e94\u5165\u5230\u6700\u63a5\u8fd1\u7684\u6574\u6570, \u8fd9\u76f8\u5f53\u4e8e\u53d6\u56fe\u50cf\u4e2d\u7684\u6bcf\u4e00\u884c\u5e76\u5c06\u5176\u5411\u4e00\u4fa7\u79fb\u52a8\u4e00\u5b9a\u7684\u91cf\u2014\u2014\u6bcf\u884c\u79fb\u52a8\u7684\u91cf\u5747\u4e0d\u540c. \u5bf9\u4e8e\u56fe\u50cf\u4e2d\u540c\u4e00\u884c\u5185\u7684\u6bcf\u4e2a\u50cf\u7d20, \u5176\u4f4d\u79fb\u91cf\u5747\u76f8\u540c, \u8fd9\u5141\u8bb8\u6211\u4eec\u5728\u7ed3\u679c\u56fe\u50cf\u4e2d\u4e0d\u80fd\u51fa\u73b0\u6d1e\u7684\u524d\u63d0\u4e0b\u8fdb\u884c\u65cb\u8f6c\u53d8\u6362. \u7ad6\u76f4\u65b9\u5411\u4e0a\u7684\u526a\u5207\u53d8\u6362\u4e5f\u8d77\u5230\u7c7b\u4f3c\u7684\u4f5c\u7528. \u5982\u6b64\u4e00\u6765, \u6211\u4eec\u4fbf\u53ef\u4ee5\u5f88\u5bb9\u6613\u5730\u5b9e\u73b0\u5149\u6805\u5316\u56fe\u50cf\u4e0a\u7684\u65cb\u8f6c\u53d8\u6362.<\/p>\n<p><strong>2. 3\u7ef4\u7ebf\u6027\u53d8\u6362<\/strong><\/p>\n<p>3\u7ef4\u7ebf\u6027\u53d8\u6362\u4e3a2\u7ef4\u7ebf\u6027\u53d8\u6362\u7684\u6269\u5c55. \u4f8b\u5982, \u6cbf\u7740\u7b1b\u5361\u5c14\u5750\u6807\u8f74\u7684\u7f29\u653e\u77e9\u9635\u4e3a$$scale(s_x, s_y, s_z) = \\begin{bmatrix}<br \/>\ns_x &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; s_y &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; s_z<br \/>\n\\end{bmatrix}.$$\u57283\u7ef4\u7a7a\u95f4\u4e2d\u8fdb\u884c\u65cb\u8f6c\u53d8\u6362\u8981\u6bd4\u57282\u7ef4\u7a7a\u95f4\u4e2d\u590d\u6742\u5f97\u591a, \u56e0\u4e3a\u6709\u66f4\u591a\u53ef\u80fd\u7684\u65cb\u8f6c\u8f74. \u7136\u800c, \u82e5\u6211\u4eec\u53ea\u662f\u60f3\u7ed5$z$\u8f74\u65cb\u8f6c, \u5219\u4ec5\u4f1a\u6539\u53d8\u5176\u7b1b\u5361\u5c14\u5750\u6807\u7684$x$\u5206\u91cf\u4e0e$y$\u5206\u91cf, \u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e0d\u5bf9\u7b1b\u5361\u5c14\u5750\u6807\u7684$z$\u5206\u91cf\u8fdb\u884c\u64cd\u4f5c\u76842\u7ef4\u65cb\u8f6c\u77e9\u9635:$$rotate-z(\\phi) = \\begin{bmatrix}<br \/>\ncos\\phi &#038; -sin\\phi &#038; 0\\\\<br \/>\nsin\\phi &#038; cos\\phi &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix}.$$\u7c7b\u4f3c\u5730, \u6211\u4eec\u53ef\u4ee5\u6784\u9020\u77e9\u9635\u6765\u56f4\u7ed5$x$\u8f74\u4e0e$y$\u8f74\u65cb\u8f6c:$$rotate-x(\\phi) = \\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; 0\\\\<br \/>\n0 &#038; cos\\phi &#038; -sin\\phi\\\\<br \/>\n1 &#038; sin\\phi &#038; cos\\phi<br \/>\n\\end{bmatrix}, \\\\ rotate-y(\\phi) = \\begin{bmatrix}<br \/>\ncos\\phi &#038; 0 &#038; sin\\phi\\\\<br \/>\n0 &#038; 1 &#038; 0\\\\<br \/>\n-sin\\phi &#038; 0 &#038; cos\\phi<br \/>\n\\end{bmatrix}.$$\u57283\u7ef4\u7a7a\u95f4\u4e2d, \u6211\u4eec\u53ef\u4ee5\u6cbf\u7740\u7279\u5b9a\u7684\u8f74\u8fdb\u884c\u526a\u5207\u53d8\u6362, \u5982$$shear-x(d_y, d_z) = \\begin{bmatrix}<br \/>\n1 &#038; d_y &#038; d_z\\\\<br \/>\n0 &#038; 1 &#038; 0\\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix}.$$\u4e0e2\u7ef4\u7ebf\u6027\u53d8\u6362\u4e00\u6837,<br \/>\n$\\\\$ $\\cdot$ \u4efb\u4f553\u7ef4\u7ebf\u6027\u53d8\u6362\u77e9\u9635\u90fd\u53ef\u4ee5\u5229\u7528SVD\u5206\u89e3\u4e3a\u65cb\u8f6c\u77e9\u9635, \u7f29\u653e\u77e9\u9635\u4e0e\u53e6\u4e00\u4e2a\u65cb\u8f6c\u77e9\u9635\u7684\u4e58\u79ef;<br \/>\n$\\\\$ $\\cdot$ \u4efb\u4f553\u7ef4\u5b9e\u5bf9\u79f0\u77e9\u9635\u90fd\u53ef\u4ee5\u5229\u7528\u7279\u5f81\u503c\u5206\u89e3\u4e3a\u65cb\u8f6c\u77e9\u9635, \u7f29\u653e\u77e9\u9635\u4e0e\u53cd\u5411\u65cb\u8f6c\u77e9\u9635\u7684\u4e58\u79ef;<br \/>\n$\\\\$ $\\cdot$ 3\u7ef4\u65cb\u8f6c\u77e9\u9635\u53ef\u4ee5\u5206\u89e3\u4e3a3\u4e2a3\u7ef4\u526a\u5207\u77e9\u9635\u7684\u4e58\u79ef.<\/p>\n<p><strong>2.1 \u4efb\u610f3\u7ef4\u65cb\u8f6c<\/strong><\/p>\n<p>3\u7ef4\u65cb\u8f6c\u77e9\u9635\u4e3a\u6b63\u4ea4\u77e9\u9635. \u4ece\u51e0\u4f55\u4e0a\u6765\u770b, \u8fd9\u610f\u5473\u7740\u77e9\u9635\u76843\u4e2a\u884c\u5411\u91cf\u662f\u76f8\u4e92\u6b63\u4ea4\u7684, 3\u4e2a\u5217\u5411\u91cf\u4ea6\u662f\u5982\u6b64. 3\u7ef4\u65cb\u8f6c\u77e9\u9635\u7684\u5f62\u5f0f\u5982\u4e0b\u6240\u793a:$$\\mathbf{R}_{\\mathbf{u} \\mathbf{v} \\mathbf{w}} = \\begin{bmatrix}<br \/>\nx_\\mathbf{u} &#038; y_\\mathbf{u} &#038; z_\\mathbf{u}\\\\<br \/>\nx_\\mathbf{v} &#038; y_\\mathbf{v} &#038; z_\\mathbf{v}\\\\<br \/>\nx_\\mathbf{w} &#038; y_\\mathbf{w} &#038; z_\\mathbf{w}<br \/>\n\\end{bmatrix}.$$\u5176\u4e2d, $\\mathbf{u} = x_\\mathbf{u} \\mathbf{x} + y_\\mathbf{u} \\mathbf{y} + z_\\mathbf{u} \\mathbf{z}$, \u540c\u7406\u53ef\u77e5$\\mathbf{v}$\u4e0e$\\mathbf{w}$\u7684\u5b9a\u4e49. \u7531\u4e8e$\\mathbf{u}$, $\\mathbf{v}$\u4e0e$\\mathbf{v}$\u8fd93\u4e2a\u884c\u5411\u91cf\u662f\u6807\u51c6\u6b63\u4ea4\u7684, \u6545$$\\mathbf{u} \\cdot \\mathbf{u} = \\mathbf{v} \\cdot \\mathbf{v} = \\mathbf{w} \\cdot \\mathbf{w} = 1, \\\\ \\mathbf{u} \\cdot \\mathbf{v} = \\mathbf{v} \\cdot \\mathbf{w} = \\mathbf{w} \\cdot \\mathbf{u} = 0.$$\u4e0a\u8ff0\u65cb\u8f6c\u77e9\u9635\u4f5c\u7528\u57283\u4e2a\u884c\u5411\u91cf\u4e0a\u5177\u6709\u7279\u6b8a\u7684\u6027\u8d28, \u5982$$\\mathbf{R}_{uvw} \\mathbf{u} = \\begin{bmatrix}<br \/>\nx_\\mathbf{u} &#038; y_\\mathbf{u} &#038; z_\\mathbf{u} \\\\<br \/>\nx_\\mathbf{v} &#038; y_\\mathbf{v} &#038; z_\\mathbf{v} \\\\<br \/>\nx_\\mathbf{w} &#038; y_\\mathbf{w} &#038; z_\\mathbf{w}<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nx_\\mathbf{u} \\\\<br \/>\ny_\\mathbf{u} \\\\<br \/>\nz_\\mathbf{u}<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\nx_\\mathbf{u} x_\\mathbf{u} + y_\\mathbf{u} y_\\mathbf{u} + z_\\mathbf{u} z_\\mathbf{u} \\\\<br \/>\nx_\\mathbf{v} x_\\mathbf{u} + y_\\mathbf{v} y_\\mathbf{u} + z_\\mathbf{v} z_\\mathbf{u} \\\\<br \/>\nx_\\mathbf{w} x_\\mathbf{u} + y_\\mathbf{w} y_\\mathbf{u} + z_\\mathbf{w} z_\\mathbf{u}<br \/>\n\\end{bmatrix} \\\\ = \\begin{bmatrix}<br \/>\n\\mathbf{u} \\cdot \\mathbf{u} \\\\<br \/>\n\\mathbf{v} \\cdot \\mathbf{u} \\\\<br \/>\n\\mathbf{w} \\cdot \\mathbf{u}<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\n1 \\\\<br \/>\n0 \\\\<br \/>\n0<br \/>\n\\end{bmatrix} = \\mathbf{x}.$$\u540c\u7406\u53ef\u5f97, $\\mathbf{R}_{uvw} \\mathbf{v} = \\mathbf{y}$, $\\mathbf{R}_{uvw} \\mathbf{w} = \\mathbf{z}$. \u6545\u901a\u8fc7\u65cb\u8f6c\u77e9\u9635$\\mathbf{R}_{uvw}$\u53ef\u5c06\u884c\u5411\u91cf$\\mathbf{u}$, $\\mathbf{v}$\u4e0e$\\mathbf{w}$\u53d8\u6362\u5230\u76f8\u5e94\u7684\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u7684\u5750\u6807\u8f74\u4e0a.<br \/>\n$\\\\$ \u6613\u77e5, $\\mathbf{R}_{uvw}^T$\u4ea6\u4e3a\u4e00\u4e2a\u6b63\u4ea4\u77e9\u9635, \u4e14$\\mathbf{R}_{uvw}^T = \\mathbf{R}_{uvw}^{-1}$(\u6b63\u4ea4\u77e9\u9635\u7684\u9006\u77e9\u9635\u603b\u4e3a\u5176\u8f6c\u7f6e\u77e9\u9635). \u91cd\u8981\u7684\u4e00\u70b9\u662f, \u5bf9\u4e8e\u65cb\u8f6c\u53d8\u6362\u77e9\u9635$\\mathbf{R}_{uvw}$, \u82e5$\\mathbf{R}_{uvw}$\u5c06\u884c\u5411\u91cf$\\mathbf{u}$\u53d8\u6362\u5230$x$\u8f74\u4e0a, \u5219$\\mathbf{R}_{uvw}^T$\u5c06$x$\u8f74\u4e0a\u7684\u5411\u91cf\u53d8\u6362\u4e3a\u4e0e\u884c\u5411\u91cf$\\mathbf{u}$\u5171\u7ebf\u7684\u5411\u91cf, \u540c\u7406\u53ef\u5f97$\\mathbf{R}_{uvw}^T$\u5206\u522b\u4f5c\u7528\u4e8e$y$\u8f74\u4e0a\u7684\u5411\u91cf\u4e0e$z$\u8f74\u4e0a\u7684\u5411\u91cf\u7684\u7ed3\u679c, \u5373$$\\mathbf{R}_{uvw}^T \\mathbf{y} = \\begin{bmatrix}<br \/>\nx_\\mathbf{u} &#038; x_\\mathbf{v} &#038; x_\\mathbf{w} \\\\<br \/>\ny_\\mathbf{u} &#038; y_\\mathbf{v} &#038; y_\\mathbf{w} \\\\<br \/>\nz_\\mathbf{u} &#038; z_\\mathbf{v} &#038; z_\\mathbf{w}<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n0 \\\\<br \/>\n1 \\\\<br \/>\n0<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\nx_\\mathbf{v} \\\\<br \/>\ny_\\mathbf{v} \\\\<br \/>\nz_\\mathbf{v}<br \/>\n\\end{bmatrix} = \\mathbf{v}.$$\u6545\u6211\u4eec\u603b\u662f\u53ef\u4ee5\u7531\u4e00\u7ec4\u6807\u51c6\u6b63\u4ea4\u57fa\u6784\u5efa\u51fa\u4e00\u4e2a\u65cb\u8f6c\u77e9\u9635.<br \/>\n$\\\\$ \u82e5\u6211\u4eec\u60f3\u5f97\u5230\u4e00\u4e2a\u5411\u91cf$\\mathbf{v}$\u7ed5\u4efb\u610f\u5411\u91cf$\\mathbf{a}$\u65cb\u8f6c\u89d2\u5ea6$\\phi$\u7684\u7ed3\u679c, \u6211\u4eec\u53ef\u4ee5\u4ee4$\\mathbf{w} = \\mathbf{a}$, \u4ece\u800c\u5f97\u5230\u4e00\u7ec4\u6807\u51c6\u6b63\u4ea4\u57fa. \u6211\u4eec\u628a\u5411\u91cf$\\mathbf{v}$\u7531\u8fd9\u7ec4\u57fa\u5411\u91cf$\\mathbf{u}$, $\\mathbf{v}$, $\\mathbf{w}$\u5bf9\u5e94\u7684\u5750\u6807\u7cfb\u53d8\u6362\u5230\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u4e0b. \u7136\u540e\u5c06\u5411\u91cf$\\mathbf{v}$\u7ed5$z$\u8f74\u65cb\u8f6c\u5f97\u5230\u5411\u91cf$\\mathbf{v}&#8217;$, \u6700\u540e\u5c06$\\mathbf{v}&#8217;$\u7531\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u53d8\u6362\u56de\u57fa\u5411\u91cf$\\mathbf{u}$, $\\mathbf{v}$, $\\mathbf{w}$\u5bf9\u5e94\u7684\u5750\u6807\u7cfb\u5373\u53ef\u5f97\u5230\u6700\u7ec8\u7684\u65cb\u8f6c\u7ed3\u679c. \u7528\u77e9\u9635\u7684\u8bed\u8a00\u5373\u53ef\u5199\u4e3a:$$\\begin{bmatrix}<br \/>\nx_\\mathbf{u} &#038; x_\\mathbf{v} &#038; x_\\mathbf{w} \\\\<br \/>\ny_\\mathbf{u} &#038; y_\\mathbf{v} &#038; y_\\mathbf{w} \\\\<br \/>\nz_\\mathbf{u} &#038; z_\\mathbf{v} &#038; z_\\mathbf{w}<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\ncos \\phi &#038; -sin \\phi &#038; 0 \\\\<br \/>\nsin \\phi &#038; cos \\phi &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nx_\\mathbf{u} &#038; y_\\mathbf{u} &#038; z_\\mathbf{u} \\\\<br \/>\nx_\\mathbf{v} &#038; y_\\mathbf{v} &#038; z_\\mathbf{v} \\\\<br \/>\nx_\\mathbf{w} &#038; y_\\mathbf{w} &#038; z_\\mathbf{w}<br \/>\n\\end{bmatrix}.$$\u6b64\u5916, \u5bf9\u4e8e\u4e00\u4e2a\u65cb\u8f6c\u77e9\u9635, \u6211\u4eec\u53ef\u4ee5\u8ba1\u7b97\u5b9e\u7279\u5f81\u503c$\\lambda = 1$(\u65cb\u8f6c\u77e9\u9635\u5fc5\u5b9a\u6709\u6b64\u7279\u5f81\u503c) \u5bf9\u5e94\u7684\u7279\u5f81\u5411\u91cf, \u8be5\u5411\u91cf\u662f\u4e0d\u53d7\u65cb\u8f6c\u5f71\u54cd\u7684.<\/p>\n<p><strong>2.2 \u6cd5\u5411\u53d8\u6362<\/strong><\/p>\n<p>\u82e5\u53d8\u6362\u77e9\u9635$\\mathbf{M}$\u4f5c\u7528\u4e8e\u66f2\u9762\u4e0a\u7684\u70b9$\\mathbf{p}$, \u90a3\u4e48\u4e8e\u70b9$\\mathbf{p}$\u5904\u4e0e\u66f2\u9762\u76f8\u5207\u7684\u5411\u91cf$\\mathbf{t}$\u5728\u7ecf\u8fc7\u53d8\u6362\u77e9\u9635$\\mathbf{M}$\u53d8\u6362\u540e\u4f9d\u65e7\u4e0e\u7ecf\u8fc7\u53d8\u6362\u77e9\u9635$\\mathbf{M}$\u53d8\u6362\u540e\u7684\u70b9$\\mathbf{p}$\u76f8\u5207. \u7136\u800c, \u70b9$\\mathbf{p}$\u5904\u7684\u66f2\u9762\u6cd5\u5411$\\mathbf{n}$\u7ecf\u8fc7\u53d8\u6362\u77e9\u9635$\\mathbf{M}$\u53d8\u6362\u540e\u53ef\u80fd\u4e0d\u5782\u76f4\u4e8e\u7ecf\u8fc7\u53d8\u6362\u77e9\u9635$\\mathbf{M}$\u53d8\u6362\u540e\u7684\u66f2\u9762.<br \/>\n$\\\\$ \u6211\u4eec\u53ef\u4ee5\u63a8\u5bfc\u51fa\u4e00\u4e2a\u53d8\u6362\u77e9\u9635$\\mathbf{N}$, s.t. \u6cd5\u5411$\\mathbf{n}$\u7ecf\u53d8\u6362\u77e9\u9635$\\mathbf{N}$\u4f5c\u7528\u540e\u4f9d\u65e7\u5782\u76f4\u4e8e\u66f2\u9762. \u6ce8\u610f\u5230, \u5bf9\u4e8e\u66f2\u9762\u4e0a\u7684\u540c\u4e00\u70b9, \u5176\u6cd5\u5411\u4e0e\u5176\u5207\u5411\u91cf\u662f\u76f8\u4e92\u5782\u76f4\u7684, \u6545\u5b83\u4eec\u7684\u70b9\u79ef\u4e3a0, \u7528\u77e9\u9635\u7684\u5f62\u5f0f\u5373\u53ef\u8868\u793a\u4e3a$$\\mathbf{n}^T \\mathbf{t} = 0.$$\u4e0d\u59a8\u4ee4$\\mathbf{t}_\\mathbf{M} = \\mathbf{M} \\mathbf{t}$, $\\mathbf{n}_\\mathbf{N} = \\mathbf{N} \\mathbf{n}$, \u6211\u4eec\u7684\u76ee\u6807\u4e3a\u627e\u5230\u6cd5\u5411$\\mathbf{n}$\u7684\u53d8\u6362\u77e9\u9635$\\mathbf{N}$, s.t. $\\mathbf{n}_\\mathbf{N}^T $$ \\mathbf{t}_\\mathbf{M} = 0$. \u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e9b\u4ee3\u6570\u6280\u5de7\u6c42\u51fa\u53d8\u6362\u77e9\u9635$\\mathbf{N}$. \u5c06$\\mathbf{M}^{\u22121} \\mathbf{M} = \\mathbf{I}$\u4ee3\u5165\u4e0a\u5f0f:$$\\mathbf{n}^T \\mathbf{t} = \\mathbf{n}^T \\mathbf{I} \\mathbf{t} = \\mathbf{n}^T \\mathbf{M}^{\u22121} \\mathbf{M} \\mathbf{t} = 0.$$\u7531\u77e9\u9635\u4e58\u6cd5\u7684\u7ed3\u5408\u5f8b\u53ef\u5f97$$(\\mathbf{n}^T \\mathbf{M}^{\u22121}) (\\mathbf{M} \\mathbf{t}) = (\\mathbf{n}^T \\mathbf{M}^{\u22121}) \\mathbf{t}_\\mathbf{M} = 0.$$\u6613\u77e5, \u884c\u5411\u91cf$\\mathbf{n}^T \\mathbf{M}^{\u22121}$\u4e0e\u5207\u5411\u91cf$\\mathbf{t}_\\mathbf{M}$\u5782\u76f4, \u5176\u4e2d, $\\mathbf{t}$\u4e3a\u66f2\u9762\u4e0a\u7684\u70b9$\\mathbf{p}$\u5904\u7684\u5207\u5e73\u9762\u4e0a\u7684\u4efb\u610f\u5207\u5411\u91cf. \u7531\u4e8e\u5728\u4e09\u7ef4\u7a7a\u95f4\u4e2d\u5782\u76f4\u4e8e\u8be5\u5207\u5e73\u9762\u7684\u6240\u6709\u5411\u91cf\u5747\u5171\u7ebf, \u53ef\u77e5$\\mathbf{n}^T \\mathbf{M}^{\u22121}$\u6216$-\\mathbf{n}^T \\mathbf{M}^{\u22121}$\u5fc5\u4e3a$\\mathbf{n}_\\mathbf{N}$\u7684\u884c\u5411\u91cf\u8868\u8fbe\u5f0f\uff0c\u5373$\\mathbf{n}_\\mathbf{N}^T$, \u6545\u4e0d\u59a8\u4ee4$$\\mathbf{n}_\\mathbf{N}^T = \\mathbf{n}^T \\mathbf{M}^{-1},$$\u5bf9\u4e0a\u5f0f\u5de6\u53f3\u4e24\u8fb9\u8f6c\u7f6e\u53ef\u5f97$$\\mathbf{n}_\\mathbf{N} = (\\mathbf{n}^T \\mathbf{M}^{-1})^T = (\\mathbf{M}^{-1})^T \\mathbf{n}.$$\u56e0\u6b64, \u6211\u4eec\u53ef\u4ee5\u770b\u5230, \u53d8\u6362\u77e9\u9635$\\mathbf{M}$\u7684\u9006\u77e9\u9635\u7684\u8f6c\u7f6e\u77e9\u9635$(\\mathbf{M}^{-1})^T$\u53ef\u4ee5\u6b63\u786e\u5730\u53d8\u6362\u6cd5\u5411$\\mathbf{n}$, \u4f7f\u5f97\u7ecf\u53d8\u6362\u540e\u7684\u5411\u91cf$\\mathbf{n}&#8217;$\u5782\u76f4\u4e8e\u7ecf\u53d8\u6362\u540e\u7684\u66f2\u9762\u4e0a\u7684\u70b9$\\mathbf{p}$\u5904\u7684\u5207\u5e73\u9762. \u7531\u4e8e\u8be5\u53d8\u6362\u77e9\u9635$\\mathbf{N}$\u53ef\u80fd\u4f1a\u6539\u53d8\u6cd5\u5411$\\mathbf{n}$\u7684\u957f\u5ea6, \u6545\u6211\u4eec\u603b\u662f\u9700\u8981\u5bf9\u53d8\u6362\u540e\u7684\u6cd5\u5411$\\mathbf{n}&#8217;$\u4f5c\u5f52\u4e00\u5316\u5904\u7406. \u6362\u53e5\u8bdd\u8bf4, \u6211\u4eec\u5e76\u4e0d\u5173\u5fc3\u53d8\u6362\u540e\u7684\u6cd5\u5411$\\mathbf{n}&#8217;$\u7684\u957f\u5ea6, \u5373\u4e0d\u9700\u8981\u7cbe\u786e\u5730\u8ba1\u7b97$\\mathbf{M}^{-1}$, \u53ea\u9700\u8981\u4fdd\u8bc1$\\mathbf{M}^{-1}$\u7684\u8fd1\u4f3c\u503c\u4e0e\u7cbe\u786e\u503c\u4e4b\u95f4\u4ec5\u76f8\u5dee\u4e00\u4e2a\u975e\u96f6\u7684\u5e38\u6570\u56e0\u5b50\u5373\u53ef. \u53c8\u77e9\u9635\u7684\u9006\u77e9\u9635\u53ef\u7531\u5176\u4f34\u968f\u77e9\u9635\u9664\u4ee5\u5176\u884c\u5217\u5f0f\u5f97\u5230, \u6545\u6211\u4eec\u53ef\u4ee5\u8df3\u8fc7\u9664\u6cd5\u8fd0\u7b97, \u5bf9\u4e8e\u4e00\u4e2a$3 \\times 3$\u7684\u53d8\u6362\u77e9\u9635$\\mathbf{M}$,$$\\mathbf{N} = \\begin{bmatrix}<br \/>\nm_{11}^c &#038; m_{12}^c &#038; m_{13}^c \\\\<br \/>\nm_{21}^c &#038; m_{22}^c &#038; m_{23}^c \\\\<br \/>\nm_{31}^c &#038; m_{32}^c &#038; m_{33}^c<br \/>\n\\end{bmatrix},$$\u5176\u4e2d, $m_{ij}$\u4e3a\u53d8\u6362\u77e9\u9635$\\mathbf{M}$\u7b2c$i$\u884c\u7b2c$j$\u5217\u7684\u5143\u7d20. \u4e0a\u5f0f\u5c55\u5f00\u5373\u5f97$$\\mathbf{N} = \\begin{bmatrix}<br \/>\nm_{22}m_{33} &#8211; m_{23}m_{32} &#038; m_{23}m_{31} &#8211; m_{21}m_{33} &#038; m_{21}m_{32} &#8211; m_{22}m_{31} \\\\<br \/>\nm_{12}m_{32} &#8211; m_{12}m_{33} &#038; m_{11}m_{33} &#8211; m_{13}m_{31} &#038; m_{12}m_{31} &#8211; m_{11}m_{32} \\\\<br \/>\nm_{12}m_{23} &#8211; m_{13}m_{22} &#038; m_{13}m_{21} &#8211; m_{11}m_{23} &#038; m_{11}m_{22} &#8211; m_{12}m_{21}<br \/>\n\\end{bmatrix}.$$<\/p>\n<p><strong>2.3 \u5e73\u79fb\u4e0e\u4eff\u5c04\u53d8\u6362<\/strong><\/p>\n<p>\u6211\u4eec\u5df2\u7ecf\u5b66\u4e60\u4e86\u7528\u77e9\u9635$\\mathbf{M}$\u53d8\u6362\u5411\u91cf\u7684\u65b9\u6cd5, \u57282\u7ef4\u7a7a\u95f4\u4e2d, \u8fd9\u4e9b\u53d8\u6362\u5f62\u5f0f\u901a\u5e38\u5982\u4e0b\u6240\u793a,$$x&#8217; = m_{11}x + m_{12}y, \\\\ y&#8217; = m_{21}x + m_{22}y.$$\u4f46\u4e0a\u8ff0\u53d8\u6362\u5f62\u5f0f\u4ec5\u80fd\u8868\u793a\u7f29\u653e\u4e0e\u65cb\u8f6c, \u65e0\u6cd5\u8868\u793a\u5e73\u79fb. \u7279\u522b\u5730, \u539f\u70b9$(0, 0)$\u5728\u7ebf\u6027\u53d8\u6362\u4e0b\u59cb\u7ec8\u4fdd\u6301\u4e0d\u52a8. \u4e3a\u4e86\u5e73\u79fb\u4e00\u4e2a\u7269\u4f53, \u5c06\u5176\u6240\u6709\u70b9\u5747\u5e73\u79fb\u76f8\u540c\u7684\u91cf, \u6211\u4eec\u9700\u8981\u4e0b\u8ff0\u53d8\u6362\u5f62\u5f0f,$$x&#8217; = x + x_t, \\\\ y&#8217; = y + y_t.$$\u4e0a\u8ff0\u53d8\u6362\u5f62\u5f0f\u65e0\u6cd5\u8868\u793a\u4e3a\u4e00\u4e2a$2 \\times 2$\u7684\u77e9\u9635\u4e58\u4ee5\u4e00\u4e2a\u5217\u5411\u91cf\u7684\u5f62\u5f0f. \u5728\u7ebf\u6027\u53d8\u6362\u7cfb\u7edf\u4e2d\u52a0\u5165\u5e73\u79fb\u7684\u4e00\u79cd\u5e38\u89c1\u6280\u5de7\u4e3a\u5c06\u4e00\u4e2a\u72ec\u7acb\u7684\u5e73\u79fb\u5411\u91cf\u4e0e\u4e00\u4e2a\u53d8\u6362\u77e9\u9635\u7ec4\u5408\u8d77\u6765, \u5176\u4e2d, \u53d8\u6362\u77e9\u9635\u4ec5\u8d1f\u8d23\u5904\u7406\u7f29\u653e\u4e0e\u65cb\u8f6c, \u800c\u5411\u91cf\u5219\u8d1f\u8d23\u5904\u7406\u5e73\u79fb. \u8fd9\u662f\u5b8c\u5168\u53ef\u884c\u7684, \u4f46\u5f62\u5f0f\u8f83\u4e3a\u7e41\u7410, \u4e14\u7ec4\u5408\u8fd9\u4e24\u4e2a\u53d8\u6362\u7684\u89c4\u5219\u5e76\u4e0d\u50cf\u7ebf\u6027\u53d8\u6362\u90a3\u6837\u7b80\u5355\u4e0e\u5e72\u51c0.<br \/>\n$\\\\$ \u76f8\u53cd\u5730, \u6211\u4eec\u53ef\u4ee5\u4f7f\u7528\u4e00\u4e2a\u806a\u660e\u7684\u6280\u5de7\u8ba9\u4e00\u4e2a\u77e9\u9635\u4e58\u6cd5\u540c\u65f6\u505a\u4e0a\u8ff0\u4e24\u4e2a\u8fd0\u7b97. \u8fd9\u4e2a\u60f3\u6cd5\u5f88\u7b80\u5355: \u7528\u4e00\u4e2a3\u7ef4\u5411\u91cf$(x, y, 1)^T$\u8868\u793a\u70b9$(x, y)$, \u5e76\u4f7f\u7528$3 \\times 3$\u7684\u77e9\u9635, \u5176\u5f62\u5f0f\u5982\u4e0b\u6240\u793a$$\\begin{bmatrix}<br \/>\nm_{11} &#038; m_{12} &#038; x_t \\\\<br \/>\nm_{21} &#038; m_{22} &#038; y_t \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix}.$$\u5176\u4e2d, \u4e0a\u8ff0\u77e9\u9635\u7684\u7b2c3\u884c\u5143\u7d20\u662f\u56fa\u5b9a\u7684, \u7528\u4e8e\u5c061\u590d\u5236\u5230\u53d8\u6362\u540e\u7684\u5411\u91cf\u4e2d, \u6545\u53d8\u6362\u540e\u7684\u6240\u6709\u5411\u91cf\u7684\u6700\u540e\u4e00\u4e2a\u5206\u91cf\u90fd\u4e3a1, \u4e14\u524d2\u884c\u8ba1\u7b97\u5411\u91cf\u7684$x$\u5206\u91cf, $y$\u5206\u91cf\u4e0e1\u7684\u7ebf\u6027\u7ec4\u5408:$$\\begin{bmatrix}<br \/>\nx&#8217; \\\\<br \/>\ny&#8217; \\\\<br \/>\n1<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\nm_{11} &#038; m_{12} &#038; x_t \\\\<br \/>\nm_{21} &#038; m_{22} &#038; y_t \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nx \\\\<br \/>\ny \\\\<br \/>\n1<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\nm_{11}x + m_{12}y + x_t \\\\<br \/>\nm_{21}x + m_{22}y + y_t \\\\<br \/>\n1<br \/>\n\\end{bmatrix}.$$\u7531\u6b64\u53ef\u89c1, \u4e0a\u8ff0\u5f0f\u5b50\u4e2d\u4ec5\u5229\u7528\u5355\u4e2a\u77e9\u9635\u4fbf\u5b9e\u73b0\u4e86\u4e00\u4e2a\u7ebf\u6027\u53d8\u6362\u4e0e\u4e00\u4e2a\u5e73\u79fb\u53d8\u6362\u7684\u7ec4\u5408\u6548\u679c. \u5176\u5bf9\u5e94\u7684\u53d8\u6362\u88ab\u79f0\u4e3a\u4eff\u5c04\u53d8\u6362, \u800c\u4e3a\u4e86\u5b9e\u73b0\u4eff\u5c04\u53d8\u6362\u589e\u52a0\u4e86\u4e00\u4e2a\u989d\u5916\u7ef4\u5ea6\u7684\u5750\u6807\u88ab\u79f0\u4e3a\u9f50\u6b21\u5750\u6807(Roberts, 1965; Riesenfeld, 1981; Penna &#038; Patterson, 1986). \u9f50\u6b21\u5750\u6807\u4e0d\u4ec5\u7b80\u5316\u4e86\u4eff\u5c04\u53d8\u6362\u7684\u5f62\u5f0f, \u800c\u4e14\u8fd8\u7b80\u5316\u4e86\u591a\u4e2a\u4eff\u5c04\u53d8\u6362\u7684\u7ec4\u5408\u5f62\u5f0f: \u4ec5\u9700\u5c06\u5bf9\u5e94\u7684\u77e9\u9635\u76f8\u4e58\u5373\u53ef.<br \/>\n$\\\\$ \u5f53\u6211\u4eec\u9700\u8981\u53d8\u6362\u8868\u793a\u65b9\u5411\u7684\u5411\u91cf\u65f6, \u4e0a\u8ff0\u5f62\u5f0f\u4fbf\u4f1a\u51fa\u73b0\u95ee\u9898\u2014\u2014\u5f53\u6211\u4eec\u5e73\u79fb\u4e00\u4e2a\u7269\u4f53\u65f6, \u8868\u793a\u65b9\u5411\u7684\u5411\u91cf\u4e0d\u5e94\u8be5\u88ab\u6539\u53d8. \u4e3a\u4e86\u89e3\u51b3\u8fd9\u4e2a\u95ee\u9898, \u6211\u4eec\u53ef\u4ee5\u5c06\u9f50\u6b21\u5750\u6807\u7684\u7b2c\u4e09\u4e2a\u5206\u91cf\u8bbe\u4e3a0:$$\\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; x_t \\\\<br \/>\n0 &#038; 1 &#038; y_t \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nx \\\\<br \/>\ny \\\\<br \/>\n0<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\nx \\\\<br \/>\ny \\\\<br \/>\n0<br \/>\n\\end{bmatrix}.$$\u9f50\u6b21\u5750\u6807\u53ef\u4ee5\u88ab\u8ba4\u4e3a\u662f\u4e00\u79cd\u5904\u7406\u5e73\u79fb\u53d8\u6362\u7684\u5de7\u5999\u601d\u60f3, \u4f46\u4ea6\u6709\u4e00\u79cd\u4e0d\u540c\u7684\u51e0\u4f55\u89e3\u91ca. \u6ce8\u610f\u5230, \u5f53\u6211\u4eec\u57fa\u4e8e\u7b1b\u5361\u5c14\u5750\u6807\u7684$z$\u5206\u91cf\u8fdb\u884c3D\u526a\u5207\u65f6, \u6211\u4eec\u4f1a\u5f97\u5230\u8fd9\u6837\u7684\u53d8\u6362:$$\\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; x_t \\\\<br \/>\n0 &#038; 1 &#038; y_t \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nx \\\\<br \/>\ny \\\\<br \/>\nz<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\nx + x_t z \\\\<br \/>\ny + y_t z \\\\<br \/>\nz<br \/>\n\\end{bmatrix}.$$\u6ce8\u610f\uff0c\u4e0a\u8ff0\u53d8\u6362\u7ed3\u679c\u5305\u542b\u4e86\u4e00\u822c\u76842D\u53d8\u6362\u5f62\u5f0f, \u4f46\u5176\u4e2d\u7684$z$\u5206\u91cf\u57282D\u4e2d\u6ca1\u6709\u4efb\u4f55\u610f\u4e49. \u63a5\u4e0b\u6765\u6211\u4eec\u4e3a\u6240\u67092D\u5750\u6807\u6dfb\u52a0\u5206\u91cf$z = 1$, \u5219\u6211\u4eec\u6709$$\\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; x_t \\\\<br \/>\n0 &#038; 1 &#038; y_t \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nx \\\\<br \/>\ny \\\\<br \/>\n1<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\nx + x_t \\\\<br \/>\ny + y_t \\\\<br \/>\n1<br \/>\n\\end{bmatrix}.$$\u901a\u8fc7\u6dfb\u52a0$(z = 1)$\u7684\u5206\u91cf\u5f97\u52302D\u5750\u6807\u7684\u9f50\u6b21\u5750\u6807, \u6211\u4eec\u73b0\u5728\u53ef\u4ee5\u5c06\u5e73\u79fb\u53d8\u6362\u5199\u4e3a\u77e9\u9635\u5f62\u5f0f. \u4f8b\u5982, \u8981\u57282D\u7a7a\u95f4\u4e2d\u5c06\u4e00\u4e2a\u70b9\u5e73\u79fb$(t_x, t_y)$, \u800c\u540e\u65cb\u8f6c\u89d2\u5ea6$\\phi$, \u6211\u4eec\u53ef\u4f7f\u7528\u53d8\u6362\u77e9\u9635$$\\mathbf{M} = \\begin{bmatrix}<br \/>\ncos \\phi &#038; -sin \\phi &#038; 0 \\\\<br \/>\nsin \\phi &#038; cos \\phi &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; x_t \\\\<br \/>\n0 &#038; 1 &#038; y_t \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix}.$$\u6ce8\u610f\u5230, 2D\u65cb\u8f6c\u77e9\u9635\u4e3a\u4e00\u4e2a$3 \\times 3$\u7684\u77e9\u9635, \u5176&#8221;\u5e73\u79fb\u69fd&#8221; \u4e2d\u542b\u67090. \u8fd9\u8868\u660e\u6211\u4eec\u662f\u4f7f\u7528\u6cbf$z = 1$\u7684\u526a\u5207\u53d8\u6362\u6765\u5904\u7406\u5e73\u79fb\u53d8\u6362\u7684. \u6211\u4eec\u53ef\u4ee5\u5c06\u4efb\u610f\u6570\u91cf\u76842D\u526a\u5207\u53d8\u6362, 2D\u65cb\u8f6c\u53d8\u6362\u4e0e2D\u5e73\u79fb\u53d8\u6362\u7684\u7ec4\u5408\u8868\u793a\u4e3a\u4e00\u4e2a\u590d\u54083D\u77e9\u9635, \u5176\u6700\u4e0b\u9762\u4e00\u884c\u603b\u4e3a$(0, 0, 1)$, \u6545\u6211\u4eec\u65e0\u9700\u5bf9\u6b64\u8fdb\u884c\u989d\u5916\u5b58\u50a8.<br \/>\n$\\\\$ \u57283D\u7a7a\u95f4\u4e2d, \u540c\u6837\u7684\u6280\u672f\u4ea6\u53ef\u7528\u4e8e\u5904\u7406\u5e73\u79fb\u53d8\u6362: \u6211\u4eec\u53ef\u4ee5\u6dfb\u52a0\u7b2c4\u4e2a\u5206\u91cf\u4ece\u800c\u5f97\u5230\u4e00\u4e2a\u9f50\u6b21\u5750\u6807, \u7136\u540e\u6211\u4eec\u8fdb\u884c\u5e73\u79fb:$$\\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; 0 &#038; x_t \\\\<br \/>\n0 &#038; 1 &#038; 0 &#038; y_t \\\\<br \/>\n0 &#038; 0 &#038; 1 &#038; z_t \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nx \\\\<br \/>\ny \\\\<br \/>\nz \\\\<br \/>\n1<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\nx + x_t \\\\<br \/>\ny + y_t \\\\<br \/>\nz + z_t \\\\<br \/>\n1<br \/>\n\\end{bmatrix}.$$\u7c7b\u4f3c\u5730, \u5bf9\u4e8e\u65b9\u5411\u5411\u91cf, \u5176\u7b2c4\u4e2a\u5206\u91cf\u4e3a0, \u6b64\u65f6\u5411\u91cf\u4e0d\u53d7\u5e73\u79fb\u53d8\u6362\u7684\u5f71\u54cd.<\/p>\n<p>\u793a\u4f8b(\u7a97\u53e3\u53d8\u6362). \u5728\u56fe\u5f62\u5b66\u4e2d, \u6211\u4eec\u7ecf\u5e38\u9700\u8981\u521b\u5efa\u4e00\u4e2a\u53d8\u6362\u77e9\u9635, \u5c06\u77e9\u5f62$[x_l, x_h] \\times $$ [y_l, y_h]$\u4e2d\u7684\u70b9\u8f6c\u6362\u81f3\u77e9\u5f62$[x_l&#8217;, x_h&#8217;] \\times [y_l&#8217;, y_h&#8217;]$\u5185. \u8fd9\u53ef\u4ee5\u901a\u8fc7\u4e00\u6b21\u7f29\u653e\u53d8\u6362\u4e0e\u5e73\u79fb\u53d8\u6362\u6765\u5b8c\u6210. \u4e0d\u8fc7, \u66f4\u6807\u51c6\u7684\u505a\u6cd5\u662f\u901a\u8fc73\u6b21\u64cd\u4f5c\u6765\u521b\u5efa\u8fd9\u4e2a\u53d8\u6362:<br \/>\n$\\\\$ (1) \u79fb\u52a8\u70b9$(x_l, y_l)$\u5230\u539f\u70b9.<br \/>\n$\\\\$ (2) \u5c06\u77e9\u5f62\u7684\u5927\u5c0f\u7f29\u653e\u81f3\u4e0e\u76ee\u6807\u77e9\u5f62\u76f8\u540c.<br \/>\n$\\\\$ (3) \u5c06\u539f\u70b9\u79fb\u52a8\u5230$(x_l&#8217;, y_l&#8217;)$.<br \/>\n$\\\\$ \u5f53\u77e9\u9635\u76f8\u4e58\u65f6, \u5728\u6700\u53f3\u8fb9\u7684\u77e9\u9635\u662f\u7b2c\u4e00\u4e2a\u4e0e\u5411\u91cf\u76f8\u4e58\u7684, \u6545\u6211\u4eec\u53ef\u4ee5\u7528\u77e9\u9635\u7684\u8bed\u8a00\u5c06\u4e0a\u8ff0\u8fc7\u7a0b\u5199\u4e3a\u5982\u4e0b\u5f62\u5f0f:$$windows = translate(x_l&#8217;, y_l&#8217;) \\ scale(\\frac{x_h&#8217; &#8211; x_l&#8217;}{x_h &#8211; x_l}, \\frac{y_h&#8217; &#8211; y_l&#8217;}{y_h &#8211; y_l}) \\ translate(-x_l, -y_l) \\\\ = \\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; x_l&#8217; \\\\<br \/>\n0 &#038; 1 &#038; y_l&#8217; \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n\\frac{x_h&#8217; &#8211; x_l&#8217;}{x_h &#8211; x_l} &#038; 0 &#038; 0 \\\\<br \/>\n0 &#038; \\frac{y_h&#8217; &#8211; y_l&#8217;}{y_h &#8211; y_l} &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; -x_l \\\\<br \/>\n0 &#038; 1 &#038; -y_l \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\\\ = \\begin{bmatrix}<br \/>\n\\frac{x_h&#8217; &#8211; x_l&#8217;}{x_h &#8211; x_l} &#038; 0 &#038; \\frac{x_l&#8217; x_h &#8211; x_h&#8217; x_l}{x_h &#8211; x_l} \\\\<br \/>\n0 &#038; \\frac{y_h&#8217; &#8211; y_l&#8217;}{y_h &#8211; y_l} &#038; \\frac{y_l&#8217; y_h &#8211; y_h&#8217; y_l}{y_h &#8211; y_l} \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix}.$$\u7c7b\u4f3c\u5730, \u6211\u4eec\u53ef\u4ee5\u5b9a\u4e49\u4e00\u4e2a3D\u7a97\u53e3\u53d8\u6362, \u5c06\u7acb\u65b9\u4f53$[x_l, x_h] \\times [y_l, y_h] \\times [z_l, z_h]$\u4e2d\u7684\u70b9\u8f6c\u6362\u81f3\u7acb\u65b9\u4f53$[x_l&#8217;, x_h&#8217;] \\times [y_l&#8217;, y_h&#8217;] \\times [z_l&#8217;, z_h&#8217;]$\u5185:$$\\begin{bmatrix}<br \/>\n\\frac{x_h&#8217; &#8211; x_l&#8217;}{x_h &#8211; x_l} &#038; 0 &#038; 0 &#038; \\frac{x_l&#8217; x_h &#8211; x_h&#8217; x_l}{x_h &#8211; x_l} \\\\<br \/>\n0 &#038; \\frac{y_h&#8217; &#8211; y_l&#8217;}{y_h &#8211; y_l} &#038; 0 &#038; \\frac{y_l&#8217; y_h &#8211; y_h&#8217; y_l}{y_h &#8211; y_l} \\\\<br \/>\n0 &#038; 0 &#038; \\frac{z_h&#8217; &#8211; z_l&#8217;}{z_h &#8211; z_l} &#038; \\frac{z_l&#8217; z_h &#8211; z_h&#8217; z_l}{z_h &#8211; z_l} \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix}.$$\u6709\u8da3\u7684\u662f, \u82e5\u6211\u4eec\u5c06\u4e00\u4e2a\u7531\u7f29\u653e\u53d8\u6362, \u526a\u5207\u53d8\u6362\u4e0e\u65cb\u8f6c\u53d8\u6362\u7ec4\u6210\u7684\u4efb\u610f\u77e9\u9635\u4e0e\u4e00\u4e2a\u7b80\u5355\u7684\u5e73\u79fb\u77e9\u9635\u76f8\u4e58, \u6211\u4eec\u53ef\u5f97$$\\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; 0 &#038; x_t \\\\<br \/>\n0 &#038; 1 &#038; 0 &#038; y_t \\\\<br \/>\n0 &#038; 0 &#038; 1 &#038; z_t \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\na_{11} &#038; a_{12} &#038; a_{13} &#038; 0 \\\\<br \/>\na_{21} &#038; a_{22} &#038; a_{23} &#038; 0 \\\\<br \/>\na_{31} &#038; a_{32} &#038; a_{33} &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\na_{11} &#038; a_{12} &#038; a_{13} &#038; x_t \\\\<br \/>\na_{21} &#038; a_{22} &#038; a_{23} &#038; y_t \\\\<br \/>\na_{31} &#038; a_{32} &#038; a_{33} &#038; z_t \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix}.$$\u56e0\u6b64, \u6211\u4eec\u53ef\u4ee5\u628a\u4efb\u610f\u53d8\u6362\u77e9\u9635\u5f7b\u5e95\u5206\u89e3\u4e3a\u7f29\u653e\/\u65cb\u8f6c\u90e8\u5206\u4e0e\u5e73\u79fb\u90e8\u5206. \u4e00\u7c7b\u91cd\u8981\u7684\u53d8\u6362\u4fbf\u662f\u521a\u4f53\u53d8\u6362. \u5b83\u4eec\u4ec5\u7531\u5e73\u79fb\u53d8\u6362\u4e0e\u65cb\u8f6c\u53d8\u6362\u7ec4\u6210, \u4e0d\u5bf9\u53d8\u6362\u5bf9\u8c61\u8fdb\u884c\u62c9\u4f38\u6216\u6536\u7f29.<\/p>\n<p><strong>2.4 \u53d8\u6362\u77e9\u9635\u7684\u9006\u77e9\u9635<\/strong><\/p>\n<p>\u6211\u4eec\u53ef\u4ee5\u4f7f\u7528SVD\u5bf9\u77e9\u9635\u8fdb\u884c\u6c42\u9006. \u7531\u4e8e\u4efb\u610f\u77e9\u9635\u5747\u53ef\u4ee5\u5206\u89e3\u4e3a\u4e00\u4e2a\u65cb\u8f6c\u77e9\u9635\u4e58\u4ee5\u4e00\u4e2a\u7f29\u653e\u77e9\u9635\u518d\u4e58\u4ee5\u4e00\u4e2a\u65cb\u8f6c\u77e9\u9635\u7684\u5f62\u5f0f, \u6545\u6c42\u9006\u5c31\u53d8\u5f97\u5341\u5206\u7b80\u5355. \u4f8b\u5982, \u57283D\u7a7a\u95f4\u4e2d\u6211\u4eec\u6709$$\\mathbf{M} = \\mathbf{R}_1 \\ scale(\\sigma_1, \\sigma_2, \\sigma_3) \\ \\mathbf{R}_2,$$\u6613\u77e5,$$\\mathbf{M}^{-1} = \\mathbf{R}_2^T \\ scale(1 \/ \\sigma_1, 1 \/ \\sigma_2, 1 \/ \\sigma_3) \\ \\mathbf{R}_1^T.$$<br \/>\n<strong>2.5 \u5750\u6807\u53d8\u6362<\/strong><\/p>\n<p>\u5f53\u6211\u4eec\u5728\u8ba1\u7b97\u673a\u4e2d\u5b58\u50a8\u6216\u4f7f\u7528\u5411\u91cf\u524d, \u6211\u4eec\u5fc5\u987b\u58f0\u660e\u5411\u91cf\u6240\u5728\u7684\u5750\u6807\u7cfb. \u4ece\u51e0\u4f55\u4e0a\u8bb2, \u4e00\u4e2a2\u7ef4\u5750\u6807\u7cfb\u7531\u4e00\u4e2a\u539f\u70b9\u4e0e\u4e00\u7ec4\u57fa(\u4e00\u4e2a2\u7ef4\u5411\u91cf\u6784\u6210\u7684\u96c6\u5408) \u7ec4\u6210. \u6211\u4eec\u901a\u5e38\u5047\u8bbe\u8fd9\u7ec4\u57fa\u662f\u4e00\u7ec4\u6807\u51c6\u6b63\u4ea4\u57fa, \u5f53\u9009\u5b9a\u7684\u5750\u6807\u7cfb\u4e3a\u7b1b\u5361\u5c14\u5750\u6807\u7cfb, \u5373\u539f\u70b9\u4e3a$\\mathbf{o}$, \u57fa\u4e3a$\\{ \\mathbf{x}, \\mathbf{y}\\}$\u65f6, 2\u7ef4\u7a7a\u95f4\u4e2d\u4e00\u4e2a\u70b9$\\mathbf{p}$\u53ef\u7531\u5750\u6807$(x_\\mathbf{p}, y_\\mathbf{p})$\u63cf\u8ff0, \u5373$$\\mathbf{p} = (x_\\mathbf{p}, y_\\mathbf{p}) \\equiv \\mathbf{o} + x_\\mathbf{p} \\mathbf{x} + y_\\mathbf{p} \\mathbf{y}.$$\u5f53\u9009\u5b9a\u7684\u5750\u6807\u7cfb\u7684\u539f\u70b9\u4e3a$\\mathbf{e}$, \u57fa\u4e3a$\\{ \\mathbf{u}, \\mathbf{v}\\}$\u65f6, 2\u7ef4\u7a7a\u95f4\u4e2d\u4e00\u4e2a\u70b9$\\mathbf{p}$\u53ef\u7531\u5750\u6807$(u_\\mathbf{p}, $$ v_\\mathbf{p})$\u63cf\u8ff0, \u5373$$p = (u_\\mathbf{p}, v_\\mathbf{p}) \\equiv \\mathbf{e} + u_\\mathbf{p} \\mathbf{u} + v_\\mathbf{p} \\mathbf{v}.$$\u6211\u4eec\u53ef\u4ee5\u7528\u77e9\u9635\u7684\u8bed\u8a00\u6765\u63cf\u8ff0\u4e0a\u8ff0\u4e24\u4e2a\u5750\u6807\u4e4b\u95f4\u7684\u5173\u7cfb, \u5982\u4e0b\u6240\u793a:$$\\begin{bmatrix}<br \/>\nx_\\mathbf{p} \\\\<br \/>\ny_\\mathbf{p} \\\\<br \/>\n1<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; x_\\mathbf{e} \\\\<br \/>\n0 &#038; 1 &#038; y_\\mathbf{e} \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nx_\\mathbf{u} &#038; x_\\mathbf{v} &#038; 0 \\\\<br \/>\ny_\\mathbf{u} &#038; y_\\mathbf{v} &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nu_\\mathbf{p} \\\\<br \/>\nv_\\mathbf{p} \\\\<br \/>\n1<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\nx_\\mathbf{u} &#038; x_\\mathbf{v} &#038; x_\\mathbf{e} \\\\<br \/>\ny_\\mathbf{u} &#038; y_\\mathbf{v} &#038; y_\\mathbf{e} \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nu_\\mathbf{p} \\\\<br \/>\nv_\\mathbf{p} \\\\<br \/>\n1<br \/>\n\\end{bmatrix}.$$\u6211\u4eec\u53ef\u5c06\u4e0a\u5f0f\u7b80\u8bb0\u4e3a$$\\mathbf{p}_{xy} = \\begin{bmatrix}<br \/>\n\\mathbf{u} &#038; \\mathbf{v} &#038; \\mathbf{e} \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\mathbf{p}_{uv}.$$\u4e0a\u5f0f\u4e2d\u7684\u53d8\u6362\u77e9\u9635\u5c06\u70b9$\\mathbf{p}$\u7531\u81ea\u5b9a\u4e49\u7684\u5750\u6807\u7cfb\u8f6c\u6362\u81f3\u7b1b\u5361\u5c14\u5750\u6807\u7cfb\u4e0b. \u53cd\u8fc7\u6765, \u6211\u4eec\u6709$$\\begin{bmatrix}<br \/>\nu_\\mathbf{p} \\\\<br \/>\nv_\\mathbf{p} \\\\<br \/>\n1<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\nx_\\mathbf{u} &#038; y_\\mathbf{u} &#038; 0 \\\\<br \/>\nx_\\mathbf{v} &#038; y_\\mathbf{v} &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; -x_\\mathbf{e} \\\\<br \/>\n0 &#038; 1 &#038; -y_\\mathbf{e} \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nx_\\mathbf{p} \\\\<br \/>\ny_\\mathbf{p} \\\\<br \/>\n1<br \/>\n\\end{bmatrix}.$$\u6211\u4eec\u53ef\u5c06\u4e0a\u5f0f\u7b80\u8bb0\u4e3a$$\\mathbf{p}_{uv} = \\begin{bmatrix}<br \/>\n\\mathbf{u} &#038; \\mathbf{v} &#038; \\mathbf{e} \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix}^{-1} \\mathbf{p}_{xy}.$$\u7c7b\u4f3c\u5730, \u6211\u4eec\u53ef\u5f973\u7ef4\u7a7a\u95f4\u4e2d\u4e0d\u540c\u5750\u6807\u7cfb\u4e0b\u7684\u5750\u6807\u4e4b\u95f4\u7684\u5173\u7cfb,$$\\begin{bmatrix}<br \/>\nx_\\mathbf{p} \\\\<br \/>\ny_\\mathbf{p} \\\\<br \/>\nz_\\mathbf{p} \\\\<br \/>\n1<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; 0 &#038; x_\\mathbf{e} \\\\<br \/>\n0 &#038; 1 &#038; 0 &#038; y_\\mathbf{e} \\\\<br \/>\n0 &#038; 0 &#038; 1 &#038; z_\\mathbf{e} \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nx_\\mathbf{u} &#038; x_\\mathbf{v} &#038; x_\\mathbf{w} &#038; 0 \\\\<br \/>\ny_\\mathbf{u} &#038; y_\\mathbf{v} &#038; y_\\mathbf{w} &#038; 0 \\\\<br \/>\nz_\\mathbf{u} &#038; z_\\mathbf{v} &#038; z_\\mathbf{w} &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nu_\\mathbf{p} \\\\<br \/>\nv_\\mathbf{p} \\\\<br \/>\nw_\\mathbf{p} \\\\<br \/>\n1<br \/>\n\\end{bmatrix}, \\\\ \\mathbf{p}_{xyz} = \\begin{bmatrix}<br \/>\n\\mathbf{u} &#038; \\mathbf{v} &#038; \\mathbf{w} &#038; \\mathbf{e} \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\mathbf{p}_{uvw},$$\u53cd\u8fc7\u6765, \u6211\u4eec\u6709$$\\begin{bmatrix}<br \/>\nu_\\mathbf{p} \\\\<br \/>\nv_\\mathbf{p} \\\\<br \/>\nw_\\mathbf{p} \\\\<br \/>\n1<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\nx_\\mathbf{u} &#038; y_\\mathbf{u} &#038; z_\\mathbf{u} &#038; 0 \\\\<br \/>\nx_\\mathbf{v} &#038; y_\\mathbf{v} &#038; z_\\mathbf{v} &#038; 0 \\\\<br \/>\nx_\\mathbf{w} &#038; y_\\mathbf{w} &#038; z_\\mathbf{w} &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; 0 &#038; -x_\\mathbf{e} \\\\<br \/>\n0 &#038; 1 &#038; 0 &#038; -y_\\mathbf{e} \\\\<br \/>\n0 &#038; 0 &#038; 1 &#038; -z_\\mathbf{e} \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\nx_\\mathbf{p} \\\\<br \/>\ny_\\mathbf{p} \\\\<br \/>\nz_\\mathbf{p} \\\\<br \/>\n1<br \/>\n\\end{bmatrix}, \\\\ \\mathbf{p}_{uvw} = \\begin{bmatrix}<br \/>\n\\mathbf{u} &#038; \\mathbf{v} &#038; \\mathbf{w} &#038; \\mathbf{e} \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix}^{-1} \\mathbf{p}_{xyz}.$$<\/p>\n<p><strong>3. \u5e38\u89c1\u95ee\u9898<\/strong><\/p>\n<p><strong>\u95ee1:<\/strong> \u80fd\u5426\u4e0d\u4f7f\u7528\u77e9\u9635\u8fdb\u884c\u53d8\u6362?<br \/>\n$\\\\$ <strong>\u7b541:<\/strong> \u53ef\u4ee5\u7684, \u4f46\u662f\u5728\u5b9e\u9645\u4f7f\u7528\u4e2d\u5c06\u66f4\u96be\u63a8\u5bfc, \u66f4\u96be\u8c03\u8bd5, \u4e14\u6548\u7387\u4e5f\u4e0d\u4f1a\u63d0\u9ad8. \u6b64\u5916, \u6240\u6709\u73b0\u4ee3\u7684\u56fe\u5f62API\u5747\u4f7f\u7528\u77e9\u9635\u8fdb\u884c\u53d8\u6362, \u56e0\u6b64\u4e3a\u4e86\u80fd\u591f\u66f4\u597d\u5730\u638c\u63e1\u5404\u79cd\u56fe\u5f62API, \u6211\u4eec\u5fc5\u987b\u7406\u89e3\u53d8\u6362\u77e9\u9635.<\/p>\n<p><strong>\u95ee2:<\/strong> \u53d8\u6362\u77e9\u9635\u7684\u6700\u540e\u4e00\u884c\u603b\u662f$(0, 0, 0, 1)$, \u5fc5\u987b\u5c06\u5176\u4fdd\u5b58\u5417?<br \/>\n$\\\\$ <strong>\u7b542:<\/strong> \u9664\u975e\u8fdb\u884c\u900f\u89c6\u53d8\u6362, \u5426\u5219\u65e0\u9700\u8fdb\u884c\u989d\u5916\u5b58\u50a8.<\/p>\n<p><strong>4. \u76f8\u5173\u4e60\u9898<\/strong><\/p>\n<p><strong>4.1<\/strong> \u5199\u51fa\u57283\u7ef4\u7a7a\u95f4\u4e2d\u5c06\u4e00\u70b9\u79fb\u52a8$(x_m, y_m, z_m)$\u7684$4 \\times 4$\u76843\u7ef4\u77e9\u9635.<br \/>\n$\\\\$ <strong>\u89e3:<\/strong> $\\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; 0 &#038; x_m \\\\<br \/>\n0 &#038; 1 &#038; 0 &#038; y_m \\\\<br \/>\n0 &#038; 0 &#038; 1 &#038; z_m \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix}$.<\/p>\n<p><strong>4.2<\/strong> \u5199\u51fa\u7ed5$y$\u8f74\u65cb\u8f6c\u89d2\u5ea6$\\theta$\u7684$4 \\times 4$\u76843\u7ef4\u77e9\u9635.<br \/>\n$\\\\$ <strong>\u89e3:<\/strong> $\\begin{bmatrix}<br \/>\ncos \\theta &#038; 0 &#038; sin \\theta &#038; 0 \\\\<br \/>\n0 &#038; 1 &#038; 0 &#038; 0 \\\\<br \/>\n-sin \\theta &#038; 0 &#038; cos \\theta &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix}$.<\/p>\n<p><strong>4.3<\/strong> \u5199\u51fa\u5728\u6240\u6709\u65b9\u5411\u4e0a\u7f29\u653e50%\u7684$4 \\times 4$\u76843\u7ef4\u77e9\u9635.<br \/>\n$\\\\$ <strong>\u89e3:<\/strong> $\\begin{bmatrix}<br \/>\n0.5 &#038; 0 &#038; 0 &#038; 0 \\\\<br \/>\n0 &#038; 0.5 &#038; 0 &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 0.5 &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix}$.<\/p>\n<p><strong>4.4<\/strong> \u5199\u51fa\u987a\u65f6\u9488\u65cb\u8f6c90\u5ea6\u76842\u7ef4\u65cb\u8f6c\u77e9\u9635.<br \/>\n$\\\\$ <strong>\u89e3:<\/strong> $\\begin{bmatrix}<br \/>\ncos \\frac{\\pi}{2} &#038; sin \\frac{\\pi}{2} \\\\<br \/>\n-sin \\frac{\\pi}{2} &#038; cos \\frac{\\pi}{2}<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\n0 &#038; 1 \\\\<br \/>\n-1 &#038; 0<br \/>\n\\end{bmatrix}.$<\/p>\n<p><strong>4.5<\/strong> \u5c064.4\u4e2d\u7684\u77e9\u9635\u5199\u4e3a3\u4e2a\u526a\u5207\u77e9\u9635\u7684\u4e58\u79ef.<br \/>\n$\\\\$ <strong>\u89e3:<\/strong> \u7531\u65cb\u8f6c\u77e9\u9635\u7684Paeth\u5206\u89e3\u53ef\u5f97$$\\begin{bmatrix}<br \/>\n0 &#038; 1 \\\\<br \/>\n-1 &#038; 0<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\ncos \\frac{\\pi}{2} &#038; sin \\frac{\\pi}{2} \\\\<br \/>\n-sin \\frac{\\pi}{2} &#038; cos \\frac{\\pi}{2}<br \/>\n\\end{bmatrix} \\\\ = \\begin{bmatrix}<br \/>\n1 &#038; \\frac{cos\\phi &#8211; 1}{sin\\phi}\\\\<br \/>\n0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n1 &#038; 0\\\\<br \/>\nsin\\phi &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n1 &#038; \\frac{cos\\phi &#8211; 1}{sin\\phi}\\\\<br \/>\n0 &#038; 1<br \/>\n\\end{bmatrix} \\\\ = \\begin{bmatrix}<br \/>\n1 &#038; -1\\\\<br \/>\n0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n1 &#038; 0\\\\<br \/>\n1 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n1 &#038; -1\\\\<br \/>\n0 &#038; 1<br \/>\n\\end{bmatrix}.$$<br \/>\n<strong>4.6<\/strong> \u8ba1\u7b97\u4e0b\u8ff0\u521a\u4f53\u53d8\u6362\u77e9\u9635\u7684\u9006\u77e9\u9635:$$\\begin{bmatrix}<br \/>\n\\mathbf{R} &#038; \\mathbf{t} \\\\<br \/>\n0 \\ 0 \\ 0  &#038; 1<br \/>\n\\end{bmatrix},$$\u5176\u4e2d, $\\mathbf{R}$\u4e3a\u4e00\u4e2a$3 \\times 3$\u7684\u65cb\u8f6c\u77e9\u9635, $\\mathbf{t}$\u4e3a\u4e00\u4e2a3\u7ef4\u5411\u91cf.<br \/>\n$\\\\$ <strong>\u89e3:<\/strong> $\\begin{bmatrix}<br \/>\n\\mathbf{R}^T &#038; -\\mathbf{t} \\\\<br \/>\n0 \\ 0 \\ 0  &#038; 1<br \/>\n\\end{bmatrix}$.<\/p>\n<p><strong>4.7<\/strong> \u8bc1\u660e\u4e00\u4e2a\u4eff\u5c04\u53d8\u6362\u7684\u77e9\u9635(\u6700\u540e\u4e00\u884c\u7684\u975e\u96f6\u5143\u4e3a\u6700\u540e\u4e00\u4e2a\u5143\u7d20, \u4e14\u4e3a1) \u7684\u9006\u77e9\u9635\u4ea6\u5177\u6709\u76f8\u540c\u7684\u5f62\u5f0f.<br \/>\n$\\\\$ <strong>\u89e3:<\/strong> \u4e00\u4e2a\u4eff\u5c04\u53d8\u6362$T: \\mathbb{R}^n \\mapsto \\mathbb{R}^n$\u53ef\u4ee5\u5199\u4e3a:$$T(\\mathbf{x}) = \\mathbf{A} \\mathbf{x} + \\mathbf{b},$$\u5176\u4e2d, $\\mathbf{x}$, $\\mathbf{b} \\in \\mathbb{R}^n$, $\\mathbf{A}$\u4e3a\u4e00\u4e2a$n \\times n$\u7684\u77e9\u9635. \u901a\u8fc7\u77e9\u9635$\\mathbf{A}$\u7684\u589e\u5e7f\u77e9\u9635, \u6211\u4eec\u53ef\u4ee5\u5f97\u5230\u4e00\u4e2a\u53d8\u6362$T&#8217;: \\mathbb{R}^{n + 1} \\mapsto \\mathbb{R}^{n + 1}$:$$T'(\\mathbf{y}) = \\mathbf{A}&#8217; \\mathbf{y},$$\u5176\u4e2d, $\\mathbf{A}&#8217; = \\begin{bmatrix}<br \/>\n\\mathbf{A} &#038; \\mathbf{b} \\\\<br \/>\n\\mathbf{0} &#038; 1<br \/>\n\\end{bmatrix}$, $\\mathbf{y} = \\begin{bmatrix}<br \/>\n\\mathbf{x} &#038; 1<br \/>\n\\end{bmatrix}^T$, $\\mathbf{0}$\u4e3a\u4e00\u4e2a$1 \\times n$\u7684\u96f6\u5411\u91cf.<br \/>\n$\\\\$ \u63a5\u4e0b\u6765, \u4e0d\u59a8\u4ee4$T(\\mathbf{x}) = \\mathbf{z}$, \u5219$\\mathbf{z} &#8211; \\mathbf{b} = \\mathbf{A} \\mathbf{x}$. \u82e5$\\mathbf{A}^{-1}$\u5b58\u5728(\u7531\u4eff\u5c04\u53d8\u6362\u5b9a\u4e49\u53ef\u77e5, \u4eff\u5c04\u53d8\u6362\u4e3a\u4e00\u4e2a\u53cc\u5c04, \u6545\u5fc5\u5b58\u5728\u9006\u53d8\u6362, \u5b8c\u6574\u5b9a\u4e49\u8be6\u89c1\u53c2\u8003\u6750\u65992), \u6211\u4eec\u6709$$\\mathbf{x} = \\mathbf{A}^{-1}(\\mathbf{z} &#8211; \\mathbf{b}) = \\mathbf{A}^{-1} \\mathbf{z} &#8211; \\mathbf{A}^{-1} \\mathbf{b},$$\u4e0a\u5f0f\u53ef\u4ee5\u6539\u5199\u4e3a$$\\mathbf{x} = \\mathbf{A}^{-1} \\mathbf{z} + \\mathbf{b}&#8217;,$$\u5176\u4e2d, $\\mathbf{b}&#8217; = -\\mathbf{A}^{-1} \\mathbf{b}$. \u8fd9\u610f\u5473\u7740\u4eff\u5c04\u53d8\u6362$T$\u7684\u9006\u53d8\u6362$T^{-1}$\u4ea6\u4e3a\u4e00\u4e2a\u4eff\u5c04\u53d8\u6362, \u4e14\u5176\u5bf9\u5e94\u7684\u589e\u5e7f\u77e9\u9635\u7684\u5206\u5757\u77e9\u9635\u5f62\u5f0f\u5982\u4e0b\u6240\u793a:$$(\\mathbf{A}^{-1})&#8217; = \\begin{bmatrix}<br \/>\n\\mathbf{A}^{-1} &#038; -\\mathbf{A}^{-1} \\mathbf{b} \\\\<br \/>\n\\mathbf{0} &#038; 1<br \/>\n\\end{bmatrix}.$$\u7efc\u4e0a\u6240\u8ff0, \u547d\u9898\u5f97\u8bc1. \u5fc5\u8981\u65f6\u53ef\u9a8c\u8bc1$\\mathbf{A}&#8217; (\\mathbf{A}^{-1})&#8217; = (\\mathbf{A}^{-1})&#8217; \\mathbf{A}&#8217; = \\mathbf{I}_{n + 1}$.<\/p>\n<p><strong>4.8<\/strong> \u63cf\u8ff0\u4e0b\u8ff02\u7ef4\u53d8\u6362\u77e9\u9635\u7684\u4f5c\u7528:$$\\begin{bmatrix}<br \/>\n0 &#038; -1 &#038; 1 \\\\<br \/>\n1 &#038; 0 &#038; 1 \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix}.$$<strong>\u89e3:<\/strong> \u9996\u5148\u5c06\u4f5c\u7528\u5bf9\u8c61\u9006\u65f6\u9488\u65cb\u8f6c90\u5ea6, \u518d\u5c06\u4f5c\u7528\u5bf9\u8c61\u5e73\u79fb$(1, 1)$.<\/p>\n<p><strong>4.9<\/strong> \u5199\u51fa\u5c06\u4e00\u4e2a2\u7ef4\u70b9$\\mathbf{x}$\u7ed5\u70b9$\\mathbf{p} = (x_\\mathbf{p}, y_\\mathbf{p})$\u65cb\u8f6c\u89d2\u5ea6$\\theta$\u7684$3 \\times 3$\u7684\u77e9\u9635.<br \/>\n$\\\\$ <strong>\u89e3:<\/strong> \u901a\u8fc73\u6b21\u64cd\u4f5c\u6765\u521b\u5efa\u8fd9\u4e2a\u53d8\u6362:<br \/>\n$\\\\$ (1) \u5c06\u70b9$\\mathbf{x}$\u4e0e\u70b9$\\mathbf{p} = (x_\\mathbf{p}, y_\\mathbf{p})$\u8fdb\u884c\u5e73\u79fb, \u4f7f\u5f97\u70b9$\\mathbf{p}$\u4e0e\u539f\u70b9\u91cd\u5408.<br \/>\n$\\\\$ (2) \u5c06\u5e73\u79fb\u540e\u7684\u70b9$\\mathbf{x}$\u7ed5\u539f\u70b9\u65cb\u8f6c\u89d2\u5ea6$\\theta$.<br \/>\n$\\\\$ (3) \u5c06\u65cb\u8f6c\u540e\u7684\u70b9$\\mathbf{x}$\u4e0e\u5e73\u79fb\u540e\u7684\u70b9$\\mathbf{p}$\u518d\u6b21\u8fdb\u884c\u5e73\u79fb, \u4f7f\u5f97\u70b9$\\mathbf{p}$\u56de\u5230\u539f\u6765\u7684\u4f4d\u7f6e.<br \/>\n$\\\\$ \u5f53\u77e9\u9635\u76f8\u4e58\u65f6, \u5728\u6700\u53f3\u8fb9\u7684\u77e9\u9635\u662f\u7b2c\u4e00\u4e2a\u4e0e\u5411\u91cf\u76f8\u4e58\u7684, \u6545\u6211\u4eec\u53ef\u4ee5\u7528\u77e9\u9635\u7684\u8bed\u8a00\u5c06\u4e0a\u8ff0\u8fc7\u7a0b\u5199\u4e3a\u5982\u4e0b\u5f62\u5f0f:$$translate(x_\\mathbf{p}, y_\\mathbf{p}) \\ rotate(\\frac{\\pi}{2}) \\ translate(-x_\\mathbf{p}, -y_\\mathbf{p}) \\\\ = \\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; x_\\mathbf{p} \\\\<br \/>\n0 &#038; 1 &#038; y_\\mathbf{p} \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\ncos \\frac{\\pi}{2} &#038; -sin \\frac{\\pi}{2} &#038; 0 \\\\<br \/>\nsin \\frac{\\pi}{2} &#038; cos \\frac{\\pi}{2} &#038; 0 \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n1 &#038; 0 &#038; -x_\\mathbf{p} \\\\<br \/>\n0 &#038; 1 &#038; -y_\\mathbf{p} \\\\<br \/>\n0 &#038; 0 &#038; 1<br \/>\n\\end{bmatrix}.$$<br \/>\n<strong>4.10<\/strong> \u5199\u51fa\u4e00\u4e2a$4 \\times 4$\u7684\u65cb\u8f6c\u77e9\u9635, \u5b83\u5c06\u4e00\u7ec43\u7ef4\u6b63\u4ea4\u5411\u91cf$$\\mathbf{u} = (x_\\mathbf{u}, y_\\mathbf{u}, z_\\mathbf{u})^T, \\\\ \\mathbf{v} = (x_\\mathbf{v}, y_\\mathbf{v}, z_\\mathbf{v})^T, \\\\ \\mathbf{w} = (x_\\mathbf{w}, y_\\mathbf{w}, z_\\mathbf{w})^T$$\u53d8\u6362\u4e3a\u53e6\u5916\u4e00\u7ec43\u7ef4\u6b63\u4ea4\u5411\u91cf$$\\mathbf{a} = (x_a, y_a, z_a)^T, \\\\ \\mathbf{b} = (x_b, y_b, z_b)^T, \\\\ \\mathbf{c} = (x_c, y_c, z_c)^T,$$\u5373$\\mathbf{M} \\mathbf{u} = \\mathbf{a}$, $\\mathbf{M} \\mathbf{v} = \\mathbf{b}$, \u4e14$\\mathbf{M} \\mathbf{w} = \\mathbf{c}$.<br \/>\n$\\\\$ <strong>\u89e3:<\/strong> \u6613\u77e5$\\mathbf{M} \\begin{bmatrix}<br \/>\n\\mathbf{u} &#038; \\mathbf{v} &#038; \\mathbf{w}<br \/>\n\\end{bmatrix} = \\begin{bmatrix}<br \/>\n\\mathbf{a} &#038; \\mathbf{b} &#038; \\mathbf{c}<br \/>\n\\end{bmatrix}$, \u4e14$\\begin{bmatrix}<br \/>\n\\mathbf{u} &#038; \\mathbf{v} &#038; \\mathbf{w}<br \/>\n\\end{bmatrix}$\u662f\u53ef\u9006\u7684, \u6545$$\\mathbf{M} = \\begin{bmatrix}<br \/>\n\\mathbf{a} &#038; \\mathbf{b} &#038; \\mathbf{c}<br \/>\n\\end{bmatrix} \\begin{bmatrix}<br \/>\n\\mathbf{u} &#038; \\mathbf{v} &#038; \\mathbf{w}<br \/>\n\\end{bmatrix}^{-1}.$$<br \/>\n<strong>4.11<\/strong> 4.10\u7b54\u6848\u4e2d\u7684\u9006\u77e9\u9635\u662f\u4ec0\u4e48?<br \/>\n$\\\\$ <strong>\u89e3:<\/strong> \u7531$\\mathbf{A}^{-1} = \\frac{\\mathbf{A}^*}{|\\mathbf{A}|}$\u53ef\u5f97$$\\begin{bmatrix}<br \/>\n\\mathbf{u} &#038; \\mathbf{v} &#038; \\mathbf{w}<br \/>\n\\end{bmatrix}^{-1} = \\frac{\\begin{bmatrix}<br \/>\ny_\\mathbf{v} z_\\mathbf{w} &#8211; y_\\mathbf{w} z_\\mathbf{v} &#038; x_\\mathbf{w} z_\\mathbf{v} &#8211; x_\\mathbf{v} z_\\mathbf{w} &#038; x_\\mathbf{v} y_\\mathbf{w} &#8211; x_\\mathbf{w} y_\\mathbf{v} \\\\<br \/>\ny_\\mathbf{w} z_\\mathbf{u} &#8211; y_\\mathbf{w} z_\\mathbf{v} &#038; x_\\mathbf{u} z_\\mathbf{w} &#8211; x_\\mathbf{w} z_\\mathbf{u} &#038; y_\\mathbf{u} x_\\mathbf{w} &#8211; x_\\mathbf{u} y_\\mathbf{w} \\\\<br \/>\ny_\\mathbf{u} z_\\mathbf{v} &#8211; y_\\mathbf{v} z_\\mathbf{u} &#038; x_\\mathbf{v} z_\\mathbf{u} &#8211; x_\\mathbf{u} z_\\mathbf{v} &#038; x_\\mathbf{u} y_\\mathbf{v} &#8211; y_\\mathbf{u} x_\\mathbf{v} \\\\<br \/>\n\\end{bmatrix}}{x_\\mathbf{u}(y_\\mathbf{v} z_\\mathbf{w} &#8211; y_\\mathbf{w} z_\\mathbf{v}) &#8211; y_\\mathbf{u}(x_\\mathbf{v} z_\\mathbf{w} &#8211; x_\\mathbf{w} z_\\mathbf{v}) + z_\\mathbf{u}(x_\\mathbf{v} y_\\mathbf{w} &#8211; x_\\mathbf{w} y_\\mathbf{v})}.$$<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u76ee\u524d\u5df2\u7ecf\u5728\u65b0\u516c\u53f8\u5de5\u4f5c\u4e86\u4e00\u5468\u591a\u4e86, \u524d\u9762\u4e24\u5929\u786e\u5b9e\u6709\u70b9\u5403\u4e0d\u6d88(\u5305\u62ec\u901a\u52e4\u65f6\u95f4\u4e0e\u5de5\u4f5c\u65f6\u95f4\u7b49\u56e0\u7d20), \u4f46\u73b0\u5728\u4e5f\u9010\u6e10\u9002\u5e94\u4e86 &hellip; <a href=\"https:\/\/www.caiqinyi.cn\/index.php\/2023\/03\/19\/transformation_matrices_mark\/\" class=\"more-link\">\u7ee7\u7eed\u9605\u8bfb<span class=\"screen-reader-text\">\u53d8\u6362\u77e9\u9635\u6ce8\u8bb0<\/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":[24],"tags":[],"_links":{"self":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3072"}],"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=3072"}],"version-history":[{"count":37,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3072\/revisions"}],"predecessor-version":[{"id":3604,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/posts\/3072\/revisions\/3604"}],"wp:attachment":[{"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/media?parent=3072"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/categories?post=3072"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.caiqinyi.cn\/index.php\/wp-json\/wp\/v2\/tags?post=3072"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}