{"id":765,"date":"2023-02-08T10:04:42","date_gmt":"2023-02-08T02:04:42","guid":{"rendered":"http:\/\/www.algmain.com\/?page_id=765"},"modified":"2023-02-13T10:11:31","modified_gmt":"2023-02-13T02:11:31","slug":"two-dimension","status":"publish","type":"page","link":"http:\/\/www.algmain.com\/index.php\/algorithm-article\/projective-variance-split\/two-dimension\/","title":{"rendered":"\u4e8c\u7ef4\u6295\u5f71\u65b9\u5dee\u5206\u5272"},"content":{"rendered":"\n

1 \u5e73\u9762\u70b9\u96c6\u7684\u6295\u5f71<\/h2>\n\n\n\n

\u4ee5\u4e0a\u8ba8\u8bba\u4e86\u4e00\u7ef4\u70b9\u96c6\u7684\u60c5\u51b5\uff0c\u4e0b\u9762\u8ba8\u8bba\u4e8c\u7ef4\u70b9\u96c6\\(A = \\{ (x_i, y_i) | i=1,2,\u2026N \\}\\)\u5728\u4efb\u4e00\u5e73\u9762\u76f4\u7ebf \\(l\\)\u4e0a\u7684\u6295\u5f71\u70b9 \\(P=\\{ p_i \u2208 |i=1,2,\u2026N \\} \\)\u7684\u8ddd\u79bb\u65b9\u5dee\u3002<\/p>\n\n\n\n

\"\"
\u56fe2 \u4e8c\u7ef4\u70b9\u96c6\u5230\u76f4\u7ebf\u7684\u6295\u5f71<\/figcaption><\/figure>\n\n\n\n

\u901a\u8fc7\u524d\u9762\u7684\u8ba8\u8bba\u53ef\u77e5\uff1a\u5206\u5272\u70b9\u5fc5\u7136\u4e3a\u6295\u5f71\u70b9\\(P=\\{ p_i \u2208 |i=1,2,\u2026N \\} \\)\u7684\u8d28\u5fc3\uff1b\u5206\u5272\u7ebf\u5fc5\u7136\u5782\u76f4\u4e8e\u76f4\u7ebf\\(l\\)\u3002\u6839\u636e\u77e2\u91cf\u6295\u5f71\u516c\u5f0f\u53ef\u77e5\uff1a<\/p>\n\n\n\n

\\[ \\overrightarrow {O{p_i}} = \\frac{{\\overrightarrow {O{A_i}} \\cdot \\overrightarrow l }}{{\\left| {\\overrightarrow l } \\right|}} \\]<\/p>\n\n\n\n

\u66f4\u8fdb\u4e00\u6b65\u6c42\u5f97\u5206\u5272\u70b9\\(p\\)\u7684\u77e2\u91cf\u4f4d\u7f6e\uff1a<\/p>\n\n\n\n

\\[\\overrightarrow {Op} = \\frac{1}{N}\\sum\\limits_{i = 1}^N {\\overrightarrow {O{p_i}} = \\frac{1}{N}} \\frac{{\\overrightarrow {O{A_i}} \\cdot \\overrightarrow l }}{{\\left| {\\overrightarrow l } \\right|}} = \\frac{{\\overrightarrow l }}{{\\left| {\\overrightarrow l } \\right|}} \\cdot \\left( {\\frac{1}{N}\\sum\\limits_{i = 1}^N {\\overrightarrow {O{A_i}} } } \\right)\\]<\/p>\n\n\n\n

\u5176\u4e2d \\(\\left( {\\frac{1}{N}\\sum\\limits_{i = 1}^N {\\overrightarrow {O{A_i}} } } \\right)\\) \u5c31\u662f\u4e8c\u7ef4\u70b9\u96c6\\(A\\)\u7684\u8d28\u5fc3\u3002\u4e5f\u5c31\u662f\u8bf4\uff1a\u5206\u5272\u70b9\u662f\u4e8c\u7ef4\u70b9\u96c6\\(A\\)\u7684\u8d28\u5fc3\u5728\u76f4\u7ebf\\(l\\)\u4e0a\u7684\u6295\u5f71\u3002<\/p>\n\n\n\n

\u8003\u8651\u66f4\u4e00\u822c\u7684\u60c5\u51b5\uff1a\u5982\u679c\u5b58\u5728\u4e00\u6761\u5e73\u884c\u4e8e\u76f4\u7ebf\\(l^\u2032\\) \u5e73\u884c\u4e8e\u76f4\u7ebf\\(l\\) \uff0c\u90a3\u4e48\u663e\u7136\u4e8c\u7ef4\u70b9\u96c6\\(A\\)\u5728\u8fd9\u4e24\u6761\u76f4\u7ebf\u4e0a\u7684\u8ddd\u79bb\u65b9\u5dee\u4e0d\u4f1a\u6709\u4efb\u4f55\u53d8\u5316\u3002\u5206\u5272\u70b9\u4ecd\u7136\u662f\u8d28\u5fc3\u5230\u76f4\u7ebf\u7684\u6295\u5f71\uff1b\u5206\u5272\u7ebf\u7684\u65b9\u5411\u4ecd\u7136\u5782\u76f4\u4e8e\u5176\u4e2d\u4efb\u4e00\u4e00\u6761\u76f4\u7ebf\u3002<\/p>\n\n\n\n

\u5728\u8fd9\u79cd\u60c5\u51b5\u4e0b\uff0c\u4e0d\u59a8\u5c06\u4e8c\u7ef4\u70b9\u96c6\\(A\\)\u7684\u8d28\u5fc3\u8bbe\u7f6e\u4e3a\u5750\u6807\u539f\u70b9\uff0c\u76f4\u7ebf\\(l\\)\u5747\u901a\u8fc7\u5750\u6807\u539f\u70b9\uff0c\u8fd9\u6837\u53ef\u4ee5\u7b80\u5316\u5206\u6790\u548c\u8ba1\u7b97\u8fc7\u7a0b\u3002<\/p>\n\n\n\n

2 \u8ddd\u79bb\u65b9\u5dee\u7684\u6781\u503c<\/h2>\n\n\n\n

\u4e09\u89d2\u51fd\u6570\u9884\u5907\u77e5\u8bc6\uff1a<\/p>\n\n\n\n

\\[\\cos 2\\alpha = 2{\\cos ^2} – 1\\]<\/p>\n\n\n\n

\\[\\sin 2\\alpha = 2\\sin \\alpha \\cos \\alpha \\]<\/p>\n\n\n\n

\\[\\sin (\\alpha \\pm \\beta ) = \\sin \\alpha \\cos \\beta \\pm \\cos \\alpha \\sin \\beta \\]<\/p>\n\n\n\n

\u4ee5\u4e8c\u7ef4\u70b9\u96c6\\(A = \\{ (x_i, y_i) | i=1,2,\u2026N \\}\\)\u7684\u8d28\u5fc3\u4e3a\u539f\u70b9\uff0c\u8bbe\u7acb\u76f4\u89d2\u5750\u6807\u7cfb\u3002\u76f4\u7ebf\\(l\\)\u8fc7\u5750\u6807\u7cfb\u539f\u70b9\uff0c\u5176\u4e0e\\(x\\)\u8f74\u7684\u5939\u89d2\u4e3a\\(\\beta\\)\u3002\u4e8c\u7ef4\u70b9\u96c6\u4e2d\u4efb\u4e00\u70b9\\(A_i = (x_i,y_i)\\)\u5230\u539f\u70b9\u7684\u8ddd\u79bb\u4e3a\\(d_i\\)\u3002\u5176\u5728\u76f4\u7ebf\\(l\\)\u4e0a\u7684\u6295\u5f71\u70b9\u5230\u539f\u70b9\u7684\u8ddd\u79bb\u4e3a\\(n_i\\) \u3002\\(A_i = (x_i,y_i)\\)\u548c\u539f\u70b9\u8fde\u7ebf\u4e0e\\(x\\)\u8f74\u7684\u5939\u89d2\u4e3a\\(n_i\\)\u3002\u663e\u7136\\(x_i=d_i cos \\alpha_i, y_i= d_i sin \\alpha_i\\)\u3002<\/p>\n\n\n\n

\"\"
\u56fe3 \u70b9\u5230\u76f4\u7ebf\u6295\u5f71\u7684\u76f8\u5173\u53c2\u6570<\/figcaption><\/figure>\n\n\n\n

\u56e0\u6b64\u6709\uff1a\\(n_i = d_i cos(\\alpha_i – \\beta)\\)\u3002\u4ee4\uff1a<\/p>\n\n\n\n

\\[H = \\sum\\limits_{i = 1}^N {n_i^2 } = \\sum\\limits_{i = 1}^N {d_i^2{{\\cos }^2}({\\alpha _i} – \\beta )} \\\\ = \\sum\\limits_{i = 1}^N {d_i^2 \\frac {\\cos 2 ( \\alpha_i – \\beta) + 1}{2}} \\\\ = \\frac{1}{2} \\sum\\limits_{i = 1}^N {d_i^2 [\\cos 2 (\\alpha_i – \\beta) + 1]} \\]<\/p>\n\n\n\n

\u73b0\u5728\u8981\u6c42\u7684\u5c31\u662f\\(H\\)\u7684\u6781\u503c\u3002\u7531\u4e8e\\(H\\)\u662f\u5173\u4e8e\\(\\beta\\)\u7684\u51fd\u6570\uff0c\u53ef\u4ee4\\(\\frac{d H}{d\\beta } = 0\\) \u3002\u5373\uff1a<\/p>\n\n\n\n

\\[\\frac{d H}{d\\beta }= \\frac{1}{2}\\sum\\limits_{i = 1}^N {d_i^2[ – \\sin 2({\\alpha _i} – \\beta )] \\cdot ( – 2)} \\\\ = \\sum\\limits_{i = 1}^N {d_i^2\\sin 2({\\alpha _i} – \\beta )} \\\\ = \\sum\\limits_{i = 1}^N {d_i^2(\\sin 2{\\alpha _i}\\cos 2\\beta – \\cos 2{\\alpha _i}\\sin 2\\beta ) = 0}\\]<\/p>\n\n\n\n

\u56e0\u6b64\u53ef\u4ee5\u6c42\u5f97\uff1a<\/p>\n\n\n\n

\\[ tg2\\beta = \\frac{\\sum\\limits_{i = 1}^N {d_i^2\\sin 2 \\alpha_i}} {\\sum\\limits_{i = 1}^N {d_i^2 \\cos 2 \\alpha _i} } = \\frac {\\sum\\limits_{i = 1}^N {d_i^2 \\cdot 2\\sin \\alpha_i \\cos \\alpha_i}} {\\sum\\limits_{i = 1}^N {d_i^2(\\cos ^2 \\alpha_i – \\sin^2 \\alpha_i)} } \\\\ = \\frac{2 \\sum \\limits_{i=1}^N {(d_i \\sin \\alpha_i) \\cdot (d_i \\cos \\alpha_i)} }{\\sum \\limits_{i=1}^N{[(d_i \\cos \\alpha_i)^2 – (d_i sin \\alpha_i)^2 ]}} = \\frac{2 \\sum \\limits_{i=1}^N{x_i y_i}}{\\sum \\limits_{i=1}^N{(x_i^2 – y_i^2)}}\\]<\/p>\n\n\n\n

\u4ee4\\(k = \\frac{2 \\sum \\limits_{i=1}^N{x_i y_i}}{\\sum \\limits_{i=1}^N{(x_i^2 – y_i^2)}}\\)\uff0c\u5219\\(tg 2 \\beta=k\\)\uff0c\\(\\beta = \\frac{1}{2} arctg(k) \\)\u3002<\/p>\n\n\n\n

\u5f53\u76f4\u7ebf\\(l\\)\u5f97\u5939\u89d2\\(\\beta = \\frac{1}{2} arctg(k) \\)\u65f6\uff0c\u8ddd\u79bb\u65b9\u5dee\u5b58\u5728\u6781\u503c\u3002\u6b64\u65f6\u662f\u6781\u5927\u503c\uff0c\u8fd8\u662f\u6781\u5c0f\u503c\u53ef\u4ee5\u5229\u7528\u4e8c\u9636\u5bfc\u6570\u8fdb\u884c\u5224\u65ad\u3002<\/p>\n\n\n\n

\\[ \\frac{d^2 H}{d^2 \\beta} = \\sum \\limits_{i=1}^N{d_i^2 cos 2 (\\alpha_i – \\beta ) \\cdot (-2) } \\\\= -2 \\sum \\limits_{i=1}^N{d_i^2 ( cos 2 \\alpha_i cos 2 \\beta + sin 2 \\alpha_i sin 2 \\beta)} \\\\ = -2 \\sum \\limits_{i=1}^N{[d_i^2 (\\cos^2 \\alpha_i – \\sin^2 \\alpha_i) \\cos 2 \\beta + 2 d_i^2 \\sin \\alpha_i \\cos \\alpha_i \\sin 2 \\beta]} \\\\ = -2 \\sum \\limits_{i=1}^N{[(x_i^2 – y_i^2) cos 2 \\beta + 2 x_i y_i sin 2 \\beta]} \\\\ = -2 [\\cos 2 \\beta \\sum \\limits_{i=1}^N{(x_i^2-y_i^2)} + 2 sin 2 \\beta \\sum \\limits_{i=1}^N{x_i y_i}] = – \\frac{4}{\\sin 2 \\beta} \\sum \\limits_{i=1}^N{x_i y_i} \\]<\/p>\n\n\n\n

\u4ece\u4ee5\u4e0a\u5206\u6790\u53ef\u77e5\uff1a\u5fc5\u7136\u5b58\u5728\u4e00\u4e2a\u8ddd\u79bb\u65b9\u5dee\u6781\u5927\u503c\u65b9\u5411\u548c\u4e00\u4e2a\u8ddd\u79bb\u65b9\u5dee\u6781\u5c0f\u503c\u65b9\u5411\uff0c\u800c\u4e14\u8fd9\u4e24\u4e2a\u65b9\u5411\u76f8\u4e92\u5782\u76f4\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

1 \u5e73\u9762\u70b9\u96c6\u7684\u6295\u5f71 \u4ee5\u4e0a\u8ba8\u8bba\u4e86\u4e00\u7ef4\u70b9\u96c6\u7684\u60c5\u51b5\uff0c\u4e0b\u9762\u8ba8\u8bba\u4e8c\u7ef4\u70b9\u96c6\u5728\u4efb\u4e00\u5e73\u9762\u76f4\u7ebf \u4e0a\u7684\u6295\u5f71\u70b9  […]<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":591,"menu_order":2,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-765","page","type-page","status-publish","hentry"],"_links":{"self":[{"href":"http:\/\/www.algmain.com\/index.php\/wp-json\/wp\/v2\/pages\/765"}],"collection":[{"href":"http:\/\/www.algmain.com\/index.php\/wp-json\/wp\/v2\/pages"}],"about":[{"href":"http:\/\/www.algmain.com\/index.php\/wp-json\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"http:\/\/www.algmain.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.algmain.com\/index.php\/wp-json\/wp\/v2\/comments?post=765"}],"version-history":[{"count":103,"href":"http:\/\/www.algmain.com\/index.php\/wp-json\/wp\/v2\/pages\/765\/revisions"}],"predecessor-version":[{"id":1517,"href":"http:\/\/www.algmain.com\/index.php\/wp-json\/wp\/v2\/pages\/765\/revisions\/1517"}],"up":[{"embeddable":true,"href":"http:\/\/www.algmain.com\/index.php\/wp-json\/wp\/v2\/pages\/591"}],"wp:attachment":[{"href":"http:\/\/www.algmain.com\/index.php\/wp-json\/wp\/v2\/media?parent=765"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}