{"id":465,"date":"2017-12-05T21:40:24","date_gmt":"2017-12-05T13:40:24","guid":{"rendered":"http:\/\/www.whudj.cn\/?p=465"},"modified":"2017-12-05T21:40:24","modified_gmt":"2017-12-05T13:40:24","slug":"b-spline%e4%ba%8c%e5%9f%ba%e5%87%bd%e6%95%b0","status":"publish","type":"post","link":"http:\/\/www.whudj.cn\/?p=465","title":{"rendered":"B-Spline(\u4e8c)\u57fa\u51fd\u6570"},"content":{"rendered":"

B\u6837\u6761\u7684\u5f62\u5f0f\u4e0eBezier\u66f2\u7ebf\u7684\u5b9a\u4e49\u65b9\u5f0f\u76f8\u540c\uff1a\\(C(u) = \\sum_{i=0}^n N_{i,n}(u)P_i \\) \u4f46\u662f\u5176\u57fa\u51fd\u6570\u7684\u5f62\u5f0f\u6709\u5f88\u5927\u7684\u53d8\u5316\u3002\u91cd\u8981\u7684\u4e00\u4e2a\u4e0d\u540c\u662f\uff0cB\u6837\u6761\u7684\u57fa\u51fd\u6570\u662f\u5b9a\u4e49\u5728\u4e00\u7ec4\u53c2\u6570\u533a\u95f4\u4e0a\u7684\u3002\u8fd9\u4e00\u8282\uff0c\u6211\u5c06\u4ecb\u7ecdB\u6837\u6761\u7684\u57fa\u51fd\u6570\u3002<\/p>\n

\u4ee4U\u4e3am+1\u4e2a\u975e\u9012\u51cf\u7684\u53c2\u6570\\(\\{u_0\\le u_1\u00a0 \\le … \\le u_m\\}\\) \u6bcf\u4e2a\\(u_i\\)\u53eb\u505a\u8282\u70b9<\/span>\uff0cU\u6210\u4e3a\u8282\u70b9\u5e8f\u5217<\/span>(Knot Vector)\uff0c\u534a\u95ed\u534a\u5f00\u533a\u95f4\\([u_i,u_{i+1})\\)\u79f0\u4e3a\u7b2ci\u4e2a\u8282\u70b9\u533a\u95f4<\/span>(knot span)\u3002\u57fa\u51fd\u6570\u7684\u8282\u70b9\u662f\u5141\u8bb8\u5b58\u5728\u91cd\u590d\u7684\uff0c\u5982\u679c\\(u_i\\)\u51fa\u73b0\u4e86k\u6b21(k>1),\u90a3\u4e48\u79f0\u5176\u4e3ak\u91cd\u8282\u70b9<\/span>\u3002\u5982\u679c\\(u_i\\)\u4ec5\u51fa\u73b0\u4e00\u6b21\uff0c\u90a3\u4e48\u79f0\u5176\u4e3a\u7b80\u5355\u8282\u70b9<\/span>\u3002\u5982\u679c\u8bf4\u6240\u6709\u7684\\([u_i,u_{i+1})\\)\u533a\u95f4\u7684\u957f\u5ea6\u90fd\u662f\u4e00\u6837\u7684\uff0c\u90a3\u4e48\u79f0\u8282\u70b9\u533a\u95f4\u4e3a\u5747\u5300<\/span>\u533a\u95f4\uff0c\u53cd\u4e4b\u79f0\u4e3a\u975e\u5747\u5300<\/span>\u533a\u95f4\u3002\u201c\u975e\u5747\u5300\u6709\u7406B\u6837\u6761\u201d\u5c31\u662f\u8bf4\u5176\u8282\u70b9\u533a\u95f4\u4e3a\u975e\u5747\u5300\u7684\u3002\u8282\u70b9\u5c06\u533a\u95f4\\([u_0,u_{m}]\\)\u5206\u6210\u591a\u6bb5\u3002\u4f46\u662f\u8003\u8651\\(N_{i,p}(u)\\)\u65f6,\u53ef\u4ee5\u8ba4\u4e3a\u5176\u5b9a\u4e49\u57df\u662f\u5728\\([u_0,u_{m}]\\)\u4e0a\u7684,\u4f46\u662f\u5176\u53ea\u5728\u6709\u9650\u7684\u533a\u95f4\u4e0a\u975e0\uff0c\u56e0\u6b64\uff0c\u5bf9\u6bd4Bezier\u66f2\u7ebf\u7684\u57fa\u51fd\u6570\uff0cB\u6837\u6761\u7684\u57fa\u51fd\u6570\u662f\u6bd4\u8f83\u5c40\u90e8\u7684\u3002<\/p>\n

B\u6837\u6761\u7684\u57fa\u51fd\u6570\u7531\u4e09\u4e2a\u53c2\u6570\u51b3\u5b9a\uff1a\u9636\u6570p\uff0c\u53c2\u6570u\uff0c\u7b2ci\u4e2a\u57fa\u51fd\u6570\u3002B\u6837\u6761\u7684\u57fa\u51fd\u6570\u7531\u9012\u5f52\u516c\u5f0f\u7ed9\u51fa\uff1a$$ N_{i,0}=\\left\\{\\begin{aligned}1&\u00a0 & u \\in [u_i,u_{i+1}) \\\\0 &\u00a0 & otherwise \\end{aligned}\\right. \\\\ N_{i,p}(u)=\\frac{u-u_i}{u_{i+p}-u_i}N_{i,p-1}(u) + \\frac{u_{i+p+1}-u}{u_{i+p+1-u_{i+1}}}N_{i+1,p-1}(u)$$<\/p>\n

\u4e0a\u8ff0\u8868\u8fbe\u5f0f\u88ab\u79f0\u4e3a Cox-de Boor \u8fed\u4ee3\u5f0f\u3002\u8fd9\u4e2a\u5f0f\u5b50\u770b\u8d77\u6765\u6bd4\u8f83\u590d\u6742\uff0c\u5b9e\u5219\u4e0d\u96be\u7406\u89e3\u3002\u4e3a\u4e86\u7406\u89e3\u65b9\u4fbf\uff0c\u6211\u4eec\u53ef\u4ee5\u4ee55\u9636B\u6837\u6761\u7684\u57fa\u51fd\u6570\u4e3a\u4f8b\uff0c\u770b\u4e00\u4e0b\\(N_{0,5}\\)\u7684\u8ba1\u7b97<\/p>\n

\"\"<\/p>\n

\\(N_{0,5}\\)\u4e3a\\(N_{0,4}\\)\uff0c\\(N_{1,4}\\)\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\\(N_{0,4}\\)\uff0c\\(N_{1,4}\\)\u53c8\u5206\u522b\u662f\\(N_{0,3}\\),\\(N_{1,3}\\),\\(N_{2,3}\\)\u7684\u7ebf\u6027\u7ec4\u5408\u2026\u2026\u7ecf\u8fc7\u4e94\u6b21\u8fed\u4ee3\u540e\uff0c\u53ef\u4ee5\u5f97\u51fa\\(N_{0,5}\\)\u4e3a5\u6b21\u591a\u9879\u5f0f\uff0c\u800c\u4e14\u5176\u5728\u533a\u95f4\\([u_0,u_6)\\)\u4e0a\u975e0\u3002<\/p>\n

\u4e0b\u56fe\u53ef\u4ee5\u5206\u6790\u4e0a\u8ff0\u7ebf\u6027\u7ec4\u5408\u7cfb\u6570\u7684\u542b\u4e49\u3002<\/p>\n

\"\"<\/p>\n

\u6839\u636edeBoor\u8fed\u4ee3\u5f0f\uff0c\\(N_{i,p}\\)\u7531\\(N_{i,p-1}\\),\\(N_{i+1,p-1}\\)\u7ec4\u5408\u800c\u6210\u3002\\(N_{i,p-1}\\)\u5728\u533a\u95f4\\([u_i,u_{i+p})\\)\u4e0a\u975e\u96f6\u3002\\(N_{i+1,p-1}\\)\u5728\u533a\u95f4\\([u_{i+1},u_{i+p+1})\\)\u4e0a\u975e\u96f6\u3002\u53ef\u4ee5\u5206\u60c5\u51b5\u8ba8\u8bba<\/p>\n

    \n
  1. \u5f53\\(u\\in [u_i,u_{i+1})\\)\u65f6\uff0c\\(N_{i+1,p-1}=0\\)\uff0c\\(N_{i,p}\\)\u4ec5\u7531\\(N_{i,p-1}\\)\u51b3\u5b9a\u3002\u7cfb\u6570\u4e3a\\(uu_i\\)\u5360\\(u_iu_{i+p}\\)\uff08\u84dd\u7ebf\uff09\u7684\u6bd4\u4f8b\u3002<\/li>\n
  2. \u5f53\u00a0\\(u\\in[u_{i+p},u_{i+p+1})\\)\u65f6\uff0c\\(N_{i,p-1}=0\\)\uff0c\\(N_{i,p}\\)\u4ec5\u7531\\(N_{i+1,p-1}\\)\u51b3\u5b9a\u3002\u7cfb\u6570\u4e3a\\(uu_{i+p+1}\\)\u5360\\(u_{i+1}u_{i+p+1}\\)\uff08\u7ea2\u7ebf\uff09\u7684\u6bd4\u4f8b\u3002<\/li>\n
  3. \u5f53\u00a0\\(u\\in[u_{i+1},u_{i+p})\\)\u65f6\uff0c\u4e24\u9879\u90fd\u4e0d\u4e3a\u96f6\uff0c\\(N_{i,p}\\)\u4e3a\u4e8c\u8005\u7684\u7ebf\u6027\u7ec4\u5408\uff0c\u7cfb\u6570\u4e0e\u4e0a\u8ff0\u76f8\u540c\u3002<\/li>\n<\/ol>\n

    \u57fa\u51fd\u6570\u7684\u6027\u8d28<\/strong><\/p>\n

    \u4e3a\u4e86\u8ba8\u8bba\u57fa\u51fd\u6570\u7684\u6027\u8d28\uff0c\u6211\u6709\u5fc5\u8981\u5148\u8d34\u51fa\u4e00\u4e2anurbs\u57fa\u51fd\u6570\u7684flash\u7684\u94fe\u63a5\uff0c\u5927\u5bb6\u53ef\u4ee5\u4e0b\u8f7d\u540e\u8bbe\u7f6e\u6253\u5f00\u65b9\u5f0f\u4e3achrome\/firefox\/IE\u81ea\u884c\u6d4f\u89c8\u3002<\/p>\n

    http:\/\/geometrie.foretnik.net\/files\/NURBS-en.swf<\/a><\/p>\n

    \u8fd9\u4e2a\u52a8\u753b\u975e\u5e38\u76f4\u89c2\u7684\u8868\u73b0\u4e86\u57fa\u51fd\u6570\u7684\u6240\u6709\u6027\u8d28\uff0c\u9700\u8981\u8ba4\u771f\u7684\u5b9e\u9a8c\u5373\u4fbf\u7ec6\u7ec6\u4f53\u4f1a\u3002<\/p>\n

      \n
    1. n\u9636B\u6837\u6761\u7684\u57fa\u51fd\u6570\u4e3an\u6b21\u591a\u9879\u5f0f\u3002\\(N_{0,5}\\)\u7684\u56fe\u91cc\u53ef\u4ee5\u770b\u5230\u3002<\/li>\n
    2. \u975e\u8d1f\u6027\u3002\u6240\u6709\u7684\\(N_{i,p} \\ge 0\\)\u3002\u5982\u4e0b\u56fe\uff0c\u53ef\u4ee5\u770b\u5230\u6240\u6709\u7684\u66f2\u7ebf\u90fd\u4f4d\u4e8e\u5750\u6807\u8f74\u4e0a\u65b9\u3002<\/li>\n
    3. \u5c40\u90e8\u6027\u3002\u5bf9\\(N_{0,5}\\)\u7684\u89c2\u5bdf\u63a8\u800c\u5e7f\u4e4b\uff0c\u53ef\u4ee5\u5f97\u51fa\\(N_{i,p}\\)\u5728\u533a\u95f4\\([u_i,u_{i+p+1})\\)\u4e0a\u975e\u96f6\u3002\u7ed3\u5408\u4e0b\u56fe\uff0c\u53ef\u4ee5\u770b\u5230\u6bcf\u4e2a\u57fa\u51fd\u6570\u4ec5\u5728\u6709\u9650\u4e2a\u533a\u95f4\u4e0a\u975e\u96f6\u3002<\/li>\n
    4. \u53cd\u8fc7\u6765\u770b\uff0c\u5728\u533a\u95f4\\([u_i,u_{i+1})\\)\u4e0a\uff0c\u975e\u96f6\u7684\u57fa\u51fd\u6570\u975e\u522b\u4e3a \\(N_{i-p},…,N_i\\)\u5171p+1\u4e2a\u3002\u5f53\u7136\uff0c\u901a\u8fc7\\(i-p \\ge 0\\)\u5f97\u51fa,\u5f53\\(i <p\\)\u65f6\uff0c\u975e\u96f6\u57fa\u51fd\u6570\u5c0f\u4e8ep+1\u4e2a\u3002\u56e0\u6b64\u5728\u533a\u95f4\\([u_i,u_{i+1})\\)\u4e0a\uff0c\u975e\u96f6\u57fa\u51fd\u6570\u81f3\u591a\u4e3ap+1\u4e2a\u3002\u7ed3\u5408\u4e0b\u56fe\u4e2d\u7684\u9ed1\u7ebf\uff0c\u53ef\u4ee5\u770b\u5230\u57280.5\u4e0e1.0\u95f4\u968f\u4fbf\u53d6\u4e00\u4e2a\u53c2\u6570u,\u5176\u5bf9\u5e94\u66f2\u7ebf\u67094\u4e2a\u975e\u96f6\u503c\u3002<\/li>\n
    5. \u5355\u4f4d\u5206\u89e3\u7387\u3002\u5728\u533a\u95f4\\([u_i,u_{i+1})\\)\u4e0a\uff0c\u975e\u96f6\u57fa\u51fd\u6570 \\(N_{i-p},…,N_i\\)\u7684\u548c\u4e3a1\u3002\u8fd9\u4e00\u70b9\u4e0ebezier\u66f2\u7ebf\u7684\u57fa\u51fd\u6570\u6027\u8d28\u76f8\u540c\uff08\u5f53\u7136\u4e86\uff0cbezier\u66f2\u7ebf\u53ea\u662fB\u6837\u6761\u7684\u4e00\u4e2a\u5b50\u96c6\uff09\u3002<\/li>\n
    6. \u5982\u679c\u8282\u70b9\u7684\u6570\u91cf\u4e3am+1\u4e2a\uff0c\u63a7\u5236\u70b9\u7684\u4e2a\u6570\u4e3an+1\u4e2a\uff0c\u5219\u5b58\u5728\u7b49\u5f0fm=n+p+1\u3002\u56e0\u4e3a\u6700\u540e\u4e00\u4e2a\u57fa\u51fd\u6570\\(N_{n,p}\\)\u5728\u533a\u95f4\\([u_n,u_{n+p+1})\\)\u4e0a\u975e\u96f6\u3002\u6240\u4ee5\u8bf4\\(u_{n+p+1}=u_m\\)\u4e3a\u6700\u540e\u4e00\u4e2a\u8282\u70b9\u3002\u8fd9\u4e2a\u7b49\u5f0f\uff0c\u6211\u4eec\u540e\u9762\u4f1a\u7ecf\u5e38\u7528\u6765\u786e\u5b9a\u8282\u70b9\u4e0e\u63a7\u5236\u70b9\u7684\u4e2a\u6570\u5173\u7cfb\u3002<\/li>\n
    7. \u57fa\u51fd\u6570\u5728\u5176\u975e\u96f6\u533a\u95f4\u4e2d\u662f\u7531\u591a\u6bb5\u591a\u9879\u5f0f\u51fd\u6570\u7ec4\u5408\u800c\u6210\u7684\u3002\u5b9e\u9645\u4e0a\uff0c\u5728\u6bcf\u4e2a\u8282\u70b9\u533a\u95f4\u4e0a\uff0c\u57fa\u51fd\u6570\u62e5\u6709\u72ec\u7acb\u7684\u591a\u9879\u5f0f\u7cfb\u6570\u3002<\/li>\n
    8. \u5982\u679c\u4e00\u4e2a\u8282\u70b9u\u7684\u91cd\u590d\u5ea6\u4e3ak,\u90a3\u4e48\u4efb\u610f\u57fa\u51fd\u6570\u5728u\u5904\u7684\u8fde\u7eed\u6027\u4e3a\\(C^{n-k}\\)\u3002<\/li>\n<\/ol>\n
      \"\"

      \u57fa\u51fd\u6570\u7684\u6027\u8d28\u8bf4\u660e<\/p><\/div>\n

      \u8282\u70b9\u591a\u91cd\u6027\u7684\u5f71\u54cd<\/p>\n

      \u5f53\u5b66\u5230\u8c03\u6574\u8282\u70b9\u533a\u95f4\u65f6\uff0c\u6211\u4eec\u5c06\u4f1a\u770b\u5230\u901a\u8fc7\u5236\u9020\u591a\u91cd\u8282\u70b9\u5e26\u6765\u7684\u7279\u6b8a\u6548\u679c\u3002\u8fd9\u91cc\uff0c\u6211\u7684\u89d2\u5ea6\u4e0eDr.Shene\u6709\u70b9\u4e0d\u540c\uff0c\u6211\u4eec\u6682\u65f6\u53ef\u4ee5\u8ba4\u4e3a\u591a\u91cd\u8282\u70b9\u5728\u57fa\u51fd\u6570\u65b9\u9762\u6ca1\u6709\u8868\u73b0\u51fa\u7279\u6b8a\u7684\u6027\u8d28\uff0c\u5373\uff1a\u957f\u5ea6\u4e3a0\u7684\u533a\u95f4\u4e5f\u662f\u6709\u6548\u7684\u533a\u95f4\u3002<\/p>\n

      \n

      Introduction to Computing with Geometry\u00a0<\/a><\/p>\n

      Architecture, geometry and computers<\/a><\/p>\n","protected":false},"excerpt":{"rendered":"

      B\u6837\u6761\u7684\u5f62\u5f0f\u4e0eBezier\u66f2\u7ebf\u7684\u5b9a\u4e49\u65b9\u5f0f\u76f8\u540c\uff1a\\(C(u) = \\sum_{i= … 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\/465"}],"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=465"}],"version-history":[{"count":19,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/465\/revisions"}],"predecessor-version":[{"id":606,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/465\/revisions\/606"}],"wp:attachment":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=465"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=465"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=465"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}