{"id":533,"date":"2017-12-12T21:54:09","date_gmt":"2017-12-12T13:54:09","guid":{"rendered":"http:\/\/www.whudj.cn\/?p=533"},"modified":"2017-12-19T15:37:27","modified_gmt":"2017-12-19T07:37:27","slug":"b-spline%e4%b8%83%e6%b1%82%e5%af%bc","status":"publish","type":"post","link":"http:\/\/www.whudj.cn\/?p=533","title":{"rendered":"B-Spline(\u4e03):\u6c42\u5bfc"},"content":{"rendered":"

\u6c42\u5bfc\u662f\u53c2\u6570\u66f2\u7ebf\u975e\u5e38\u6700\u91cd\u8981\u7684\u529f\u80fd\u4e4b\u4e00\u3002\u6bd4\u5982\u5207\u7ebf\u8ba1\u7b97\u3001\u66f2\u7ebf\u62df\u5408\u7b49\u7b49\u3002\u672c\u8282\u7684\u4e3b\u8981\u76ee\u7684\u662f\u4ecb\u7ecdB\u6837\u6761\u66f2\u7ebf\u6c42\u5bfc\u7684\u65b9\u6cd5\u3002(ps:\u56e0\u4e3a\u5185\u5bb9\u91cc\u542b\u6709\u77e9\u9635\uff0c\u5982\u679c\u6ca1\u6709\u6b63\u786e\u663e\u793a\u8bf7\u5237\u65b0\u4e00\u4e0b)\u3002<\/p>\n

B\u6837\u6761\u57fa\u51fd\u6570\u7684\u5bfc\u6570\u662f:$$\\frac{d}{du}N_{i,p}(u)=N_{i,p}^\\prime (u)=\\frac{p}{u_{i+p}-u_i}N_{i,p-1}-\\frac{p}{u_{i+p+1}-u_{i+1}}N_{i+1,p-1}$$\u8fd9\u4e2a\u516c\u5f0f\u5e94\u8be5\u662f\u4ecede-Boor\u9012\u63a8\u63a8\u5bfc\u800c\u6765\u7684\uff0c\u6211\u76f4\u63a5\u5f15\u7528\u7684Dr.shene\u7684\u8bb2\u4e49<\/a>\uff0c\u4ed6\u540c\u6837\u6ca1\u6709\u7ed9\u51fa\u8be6\u7ec6\u63a8\u5bfc\u8fc7\u7a0b\u3002\u8fd9\u4e2a\u63a8\u5bfc\u597d\u5904\u5c31\u662f\u5c06p\u9636\u57fa\u51fd\u6570\u7684\u5bfc\u6570\u7528p-1\u9636\u57fa\u51fd\u6570\u7684\u65b9\u5f0f\u8868\u73b0\u51fa\u6765\uff0c\u53bb\u6389\u4e86\u4f4e\u9636\u57fa\u51fd\u6570\u4e0a\u7684\u5bfc\u6570\u7b26\u53f7\u3002\u7528B\u6837\u6761\u7684\u77e9\u9635\u5f62\u5f0f\u8868\u8fbeB\u6837\u6761\u7684\u5bfc\u6570\uff1a$$ C^\\prime (u)= \\begin{bmatrix} N_0&N_1&…&N_n&N_{n+1} \\end{bmatrix}\\begin{bmatrix}a_0P_0 \\\\ a_1P_1 \\\\ … \\\\ a_nP_n \\\\ 0\\end{bmatrix}\u00a0 \u00a0+ \\begin{bmatrix} N_0&N_1&…&N_n&N_{n+1} \\end{bmatrix}\\begin{bmatrix}0 \\\\ -a_1P_0 \\\\ … \\\\ -a_{n}P_{n-1} \\\\ -a_{n+1}P_n\\end{bmatrix} \\\\ =\\begin{bmatrix} N_0&N_1&…&N_n&N_{n+1} \\end{bmatrix}\\begin{bmatrix}a_0P_0 \\\\ a_1(P_1-P_0) \\\\ … \\\\a_n(p_n-p_{n-1}) \\\\ -a_{n+1}P_n \\end{bmatrix} $$<\/p>\n

\u5176\u4e2d \\(a_i = \\frac{p}{u_{i+p}-u_i}\\)\u3002\u800c\u4e14\uff0c\u4e3a\u4e86\u8868\u8fbe\u7b80\u4fbf\uff0c\u5c06\\(N_{i,p-1}\\)\u7b80\u5199\u4e3a\\(N_i\\)\u3002\u9700\u8981\u6ce8\u610f\u7684\u662f,\u5bf9\u4e8e\u6700\u5e38\u7528\u7684\u201cclamped B Spline\u201d,\\(a_0=a_{n+1}=\\frac{p}{0}\\),\u6839\u636eDr.shene\u7ed9\u51fa\u7684\u7ed3\u8bba\uff0c\u663e\u7136\u8fd9\u4e24\u9879\u88ab\u5f53\u505a0\u5904\u7406\u4e86\u3002\u56e0\u6b64B\u6837\u6761\u7684\u5bfc\u6570\u7684\u6700\u7ec8\u5f62\u5f0f\u662f\uff1a$$C^\\prime (u)=\\begin{bmatrix}N_1,…,N_n\\end{bmatrix}\\begin{bmatrix} a_1(P_1-P_0) \\\\ … \\\\a_n(p_n-p_{n-1}) \\end{bmatrix}$$$$a_i = \\frac{p}{u_{i+p}-u_i}$$<\/p>\n

\u8fd9\u8bf4\u660ep\u9636\u201cClamped\u201dB\u6837\u6761\u7684\u5bfc\u6570\u662fp-1\u9636B\u6837\u6761\uff0c\u800c\u4e14\u5bfc\u6570\u7684\u63a7\u5236\u70b9\u4e2a\u6570\u6bd4B\u6837\u6761\u5c11\u4e00\u4e2a\uff0c\u8282\u70b9\u4e2a\u6570\u6bd4B\u6837\u6761\u5c112\u4e2a\uff0c\u5206\u522b\u53bb\u6389\u4e86\u539f\u8282\u70b9\u5e8f\u5217\u7684\u9996\u5c3e\u8282\u70b9\uff0c\u6240\u4ee5\u5bfc\u6570\u66f2\u7ebf\u4e5f\u662f\u201cclamped\u201dB\u6837\u6761\u3002<\/strong><\/p>\n

\u4f8b\u5982\uff0c3\u9636B\u6837\u6761\u7684\u8282\u70b9\u5e8f\u5217\u662f\\(\\{0,0,0,0,…,t,t,t,t\\}\\)\u90a3\u4e48\u5176\u5bfc\u6570\u66f2\u7ebf\u7684\u8282\u70b9\u5e8f\u5217\u5c31\u662f\\(\\{0,0,0,…,t,t,t\\}\\)\u3002<\/p>\n

\u6b32\u6c42\u66f2\u7ebf\u5728\u53c2\u6570u\u5904\u7684\u5bfc\u6570\uff0c\u53ea\u9700\u8981\u8ba1\u7b97\u51faB\u6837\u6761\u5bfc\u6570\u6837\u6761\u66f2\u7ebf\u7684\u63a7\u5236\u70b9\u5e8f\u5217\uff0c\u7136\u540e\u5e94\u7528de Boor\u7b97\u6cd5\u6c42\u5bfc\u6570\u66f2\u7ebf\u7684\u503c\u5c31\u53ef\u4ee5\u4e86\u3002\u6240\u4ee5\u8bf4\u6837\u6761\u66f2\u7ebf\u7684\u6c42\u5bfc\u5341\u5206\u65b9\u4fbf\u3002<\/p>\n

\u518d\u6765\u770b\u201cclamped\u201dB\u6837\u6761\u9996\u5c3e\u70b9\u5207\u7ebf\u7684\u60c5\u51b5\u3002\u901a\u8fc7\u5c06\u9996\u5c3e\u8282\u70b9\u7684\u91cd\u590d\u5ea6\u53d8\u4e3ap+1\uff0c\u6211\u4eec\u5df2\u7ecf\u83b7\u5f97\u4e86\u4e0eBezier\u66f2\u7ebf\u4e00\u6837\u826f\u597d\u7684\u6027\u8d28\uff1a\u9996\u5c3e\u70b9\u5206\u522b\u4e0e\u9996\u5c3e\u63a7\u5236\u70b9\u76f8\u540c\u3002\u800c\u9996\u5c3e\u70b9\u7684\u5207\u5411\u91cf\uff0c\u6b63\u662f\\(a_1(P_1-P_0)\\)\u4e0e\\(a_n(P_n-P_{n-1})\\)\uff0c\u8bf4\u660eB\u6837\u6761\u9996\u5c3e\u5904\u7684\u4e14\u5411\u91cf\u4e0e\u63a7\u5236\u591a\u8fb9\u5f62\u7684\u9996\u672b\u76f8\u5207\uff0c\u4e5f\u4e0eBezier\u7684\u6027\u8d28\u4e00\u6837\u3002<\/p>\n

\u6240\u4ee5\u8bf4\uff0c\u4ece\u5e94\u7528\u7684\u89d2\u5ea6\u8bb2\uff0c\u56e0\u4e3a\u5229\u4e8e\u4f7f\u7528\uff0c\u65b9\u4fbf\u5b9e\u73b0\uff0c\u201cclamped\u201dB\u6837\u6761\u7684\u4f7f\u7528\u573a\u666f\u5927\u5927\u8d85\u8fc7\u4e86\u975e\u201cclamped\u201d\u60c5\u51b5\u3002\u56e0\u6b64\uff0c\u6211\u540e\u9762\u6709\u5173B-Spline\u8bdd\u9898\uff0c\u90fd\u4f1a\u7740\u91cd\u8bb2\u5230\u201cclamped\u201dB\u6837\u6761\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

\u6c42\u5bfc\u662f\u53c2\u6570\u66f2\u7ebf\u975e\u5e38\u6700\u91cd\u8981\u7684\u529f\u80fd\u4e4b\u4e00\u3002\u6bd4\u5982\u5207\u7ebf\u8ba1\u7b97\u3001\u66f2\u7ebf\u62df\u5408\u7b49\u7b49\u3002\u672c\u8282\u7684\u4e3b\u8981\u76ee\u7684\u662f … Continue reading →<\/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":[23],"tags":[],"_links":{"self":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/533"}],"collection":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=533"}],"version-history":[{"count":19,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/533\/revisions"}],"predecessor-version":[{"id":613,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/533\/revisions\/613"}],"wp:attachment":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=533"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=533"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=533"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}