{"id":493,"date":"2017-12-07T21:38:35","date_gmt":"2017-12-07T13:38:35","guid":{"rendered":"http:\/\/www.whudj.cn\/?p=493"},"modified":"2017-12-07T21:38:41","modified_gmt":"2017-12-07T13:38:41","slug":"b-spline%e4%b8%89%e6%a0%b7%e6%9d%a1%e6%9b%b2%e7%ba%bf%e7%9a%84%e6%80%a7%e8%b4%a8","status":"publish","type":"post","link":"http:\/\/www.whudj.cn\/?p=493","title":{"rendered":"B-Spline(\u4e09)\u6837\u6761\u66f2\u7ebf\u7684\u6027\u8d28"},"content":{"rendered":"

\u672c\u8282\u7684\u76ee\u6807\u4e3a\u4ecb\u7ecdB\u6837\u6761\u66f2\u7ebf\u7684\u5b9a\u4e49\uff0c\u4ee5\u53ca\u5176\u91cd\u8981\u6027\u8d28\u3002\u8fd9\u4e9b\u6027\u8d28\uff0c\u5bf9\u6837\u6761\u66f2\u7ebf\u7684\u5e94\u7528\u8d77\u4e86\u51b3\u5b9a\u6027\u7684\u4f5c\u7528\u3002<\/p>\n

1.\u5b9a\u4e49:\u00a0<\/strong><\/p>\n

\u4e0eBezier\u66f2\u7ebf\u7684\u5b9a\u4e49\u76f8\u4f3c\uff0cB\u6837\u6761\u66f2\u7ebf\u7684\u5b9a\u4e49\u662f\uff1a\u7ed9\u5b9an+1\u4e2a\u63a7\u5236\u70b9\\(\\{P_0,P_1,…,P_n\\}\\),m+1\u4e2a\u8282\u70b9\\(\\{u_0,u_1,…,u_m\\}\\),p\u9636B\u6837\u6761\u66f2\u7ebf\\(C(u)=\\sum_{i=0}^n N_{i,p}(u)P_i\\)\u3002\u76f8\u5bf9Bezier\u66f2\u7ebf\u4e0d\u540c\u4e4b\u5904\u5728\u4e8e\uff1a<\/p>\n

    \n
  1. B\u6837\u6761 \u7684\u63a7\u5236\u70b9\u4e2a\u6570\u4e0e\u66f2\u7ebf\u6b21\u6570\u65e0\u5173\uff0c\u56e0\u6b64B\u6837\u6761\u7684\u81ea\u7531\u5ea6\u66f4\u5927\uff0c\u53ef\u4ee5\u5b9a\u4e49\u5f88\u591a\u63a7\u5236\u70b9\u53c8\u4e0d\u7528\u62c5\u5fc3\u66f2\u7ebf\u6b21\u6570\u8fc7\u9ad8\u800c\u8ba1\u7b97\u56f0\u96be\u3002<\/li>\n
  2. B\u6837\u6761\u6709\u591a\u4e2a\u8282\u70b9\u533a\u95f4\uff0c\u4f7f\u5f97\u5176\u4e3a\u5206\u6bb5\u51fd\u6570\uff0c\u800cBezier\u66f2\u7ebf\u53ea\u6709\u4e00\u4e2a\u533a\u95f4\u3002<\/li>\n<\/ol>\n

    \u4e8c\u8005\u7684\u8054\u7cfb\u662f\uff1a<\/p>\n

      \n
    1. B\u6837\u6761\u4e0a\u4efb\u610f\u4e00\u4e2a\u533a\u95f4\\([u_i,u_{i+1})\\)\u4e0a\u7684\u66f2\u7ebf\u7684\u5b9a\u4e49\u90fd\u662f\u4e00\u4e2aBezier\u3002\u56e0\u6b64\u53ef\u4ee5\u5c06B\u6837\u6761\u8f6c\u4e3a\u5206\u6bb5bezier\u66f2\u7ebf\u3002<\/li>\n
    2. Bezier\u662f\u4e00\u79cd\u7279\u6b8a\u7684\u6837\u6761\u66f2\u7ebf\u3002\u540e\u9762\u4f1a\u8be6\u7ec6\u5206\u6790\u3002<\/li>\n<\/ol>\n

      2.\u6027\u8d28<\/strong><\/p>\n

      \u56e0\u4e3a\u662fbezier\u66f2\u7ebf\u7684\u4e00\u822c\u5f62\u5f0f\uff0cB\u6837\u6761\u66f2\u7ebf\u7684\u5f88\u591a\u6027\u8d28\u90fd\u548cbezier\u66f2\u7ebf\u76f8\u540c\u3002\u800c\u4e14\uff0cB\u6837\u6761\u6709\u5f88\u591a\u66f4\u597d\u7684\u6027\u8d28\u3002\u4e00\u6761\u7531n+1\u4e2a\u63a7\u5236\u70b9\uff0cm+1\u4e2a\u8282\u70b9\uff0cp\u6b21(m=n+p+1)\u7684B\u6837\u6761\u66f2\u7ebf\u6709\u5982\u4e0b\u6027\u8d28\uff1a<\/p>\n

        \n
      1. B\u6837\u6761\u4e0a\u4efb\u610f\u4e00\u4e2a\u533a\u95f4\\([u_i,u_{i+1})\\)\u4e0a\u7684\u66f2\u7ebf\u7684\u5b9a\u4e49\u90fd\u662f\u4e00\u4e2aBezier\u66f2\u7ebf\u3002\u76f8\u5bf9\u4e8ebezier\u66f2\u7ebf\uff0cB\u6837\u6761\u66f2\u7ebf\u80fd\u591f\u66f4\u597d\u7684\u63a7\u5236\u66f2\u7ebf\u5f62\u72b6\u3002\u5982\u4e0b\u56fe\uff0c\u5de6\u4fa7\u662f11\u4e2a\u63a7\u5236\u70b9\u6784\u6210\u76843\u6b21\u6837\u6761\u66f2\u7ebf\uff0c\u53f3\u4fa7\u662f\u540c\u6837\u63a7\u5236\u70b9\u6784\u6210\u768410\u6b21Bezier\u66f2\u7ebf\uff0c\u5176\u66f2\u7ebf\u53ef\u63a7\u5236\u6027\u8981\u6bd4\u6837\u6761\u66f2\u7ebf\u5dee\u3002\u5982\u679c\u5b9a\u4e49\u4e00\u6837\u7684\u63a7\u5236\u70b9\uff0c\u9636\u6b21\u8d8a\u4f4e\uff0c\u66f2\u7ebf\u79bb\u63a7\u5236\u70b9\u5c31\u8d8a\u8fdc\u3002\u5f53\u7136\u4e86\uff0c\u5982\u679c\u66f2\u7ebf\u7684\u9636\u6b21\u4e3a1\uff0c\u90a3\u4e48\u66f2\u7ebf\u5c31\u662f\u63a7\u5236\u591a\u8fb9\u5f62\u672c\u8eab==!\u3002\"\"<\/li>\n
      2. m,n,p\u5fc5\u987b\u8981\u6ee1\u8db3\u7684\u5173\u7cfb\u662fm=n+p+1\u3002\u8fd9\u4e2a\u5728\u57fa\u51fd\u6570\u7684\u5b9a\u4e49\u91cc\u9762\u5c31\u8bc1\u660e\u4e86\u3002<\/li>\n
      3. \u201cclamped\u201dB\u6837\u6761\uff08\u5982\u4e0a\u56fe\u5de6\uff09\uff0c\u5c31\u662f\u9996\u5c3e\u8282\u70b9\u91cd\u590d\u5ea6\u5206\u522b\u4e3ap+1\u7684\u6837\u6761\uff0c\u66f2\u7ebf\u7684\u9996\u5c3e\u70b9\u5206\u522b\u548c\u9996\u5c3e\u63a7\u5236\u70b9\u76f8\u540c\u3002\u8fd9\u4e2a\u548cbezier\u66f2\u7ebf\u4e00\u6837\u3002\u5982\u4f55\u8bc1\u660e\u5462\uff1f\u53ef\u4ee5\u5c06u=0\u5e26\u5165de-Boor\u8fed\u4ee3\u5f0f\u4e2d\u6c42\u89e3\uff0c\u53ea\u4e0d\u8fc7\uff0c\u56e0\u4e3a\u91cd\u590d\u8282\u70b9(0\u957f\u5ea6\u533a\u95f4)\u7684\u5b58\u5728\uff0c\u6211\u4eec\u9700\u8981\u5b9a\u4e49\\(\\frac{0}{0}=0\\)\u3002\u5b9e\u9645\u4e0a\uff0c\u53ef\u4ee5\u901a\u8fc7\u4e0a\u8282\u6211\u4ecb\u7ecd\u7684swf\u6587\u4ef6\u76f4\u63a5\u89c2\u5bdfB\u6837\u6761\u57fa\u51fd\u6570\u7684\u60c5\u51b5\u3002\u5982\u4e0b\u56fe\uff0c\\(N_0,N_6\\)\u5728\u8282\u70b9\u533a\u95f4\u9996\u672b\u70b9\u5904\u5747\u4e3a1\uff0c\u5176\u4ed6\u57fa\u51fd\u6570\u4e3a0\uff0c\u90a3\u4e48\u5e26\u5165B\u6837\u6761\uff0c\u53ef\u4ee5\u5f97\u51fa\\(C(0)=P_0;C(1)=P_6\\)\u3002\"\"<\/li>\n
      4. \u5f3a\u51f8\u5305\u6027:\u5bf9\u5e94\u533a\u95f4\\([u_i,u_{i+1})\\)\u4e0a\u66f2\u7ebf\u4e00\u5b9a\u5728\u7531p+1\u4e2a\u63a7\u5236\u70b9\\(P_{i-p},…,P{i}\\)\u6784\u6210\u7684\u51f8\u5305\u4e4b\u4e2d\u3002\u56e0\u4e3a\u5728\u8fd9\u6bb5\u533a\u95f4\u4e0a\uff0c\\(N_{i-p},…,N{i}\\)\u662f\u5927\u4e8e0\u7684\uff08\u539f\u6587\u4e3a\u975e0\uff09\uff0c\u800c\u4e14\u6839\u636e\u57fa\u51fd\u6570\u7684\u6027\u8d28\uff0c\u51e0\u8005\u76f8\u52a0\u7b49\u4e8e1\uff0c\u90a3\u4e48\u4e2d\u95f4\u7684\u66f2\u7ebf\u76f8\u5f53\u4e8e\u8fd9\u51e0\u4e2a\u70b9\u7684\u201c\u52a0\u6743\u5e73\u5747\u201d\uff0c\u81ea\u7136\u4f4d\u4e8e\u5176\u51f8\u5305\u4e4b\u5185\u3002\u6240\u8c13\u5f3a\u51f8\u5305\u6027\uff0c\u6307\u7684\u662f\u66f2\u7ebf\u8981\u6bd4\u51f8\u5305\u8981\u5c0f\u7684\u591a\u3002\u8fd9\u4e2a\u6027\u8d28\u5728\u66f2\u7ebf\u7684\u6298\u7ebf\u5316\uff08\u79bb\u6563\uff09\u65f6\u975e\u5e38\u6709\u7528\u3002\u5982\u679c\u6211\u4eec\u80fd\u8ba9\u63a7\u5236\u70b9\u6784\u6210\u76f4\u7ebf\uff0c\u90a3\u4e48\u5bf9\u5e94\u7684\u66f2\u7ebf\u4e5f\u80fd\u5f62\u6210\u76f4\u7ebf\u3002\"\"<\/li>\n
      5. \u5c40\u90e8\u4fee\u6539\u6a21\u5f0f\uff1a\u4fee\u6539\u63a7\u5236\u70b9\\(P_i\\)\uff0c\u90a3\u4e48\u5176\u5f71\u54cd\u7684\u66f2\u7ebf\u8303\u56f4\u4ec5\u9650\u4e8e\u533a\u95f4\\([u_i,u_{i+p+1}\\)\u3002\u5bf9\u5e94\u7684\u662f\u57fa\u51fd\u6570\u7684\u975e\u96f6\u8303\u56f4\u3002\u56e0\u6b64\uff0c\u5927\u5bb6\u53ef\u4ee5\u8bb0\u4f4f\uff0c\\(N_i\\)\u4e0e\\(P_i\\)\u662f\u4e00\u5bf9\uff0c\u5982\u679c\\(N_i\\)\u975e\u96f6\uff0c\u90a3\u4e48\\(P_i\\)\u7684\u53d8\u5316\u5c31\u4f1a\u5bf9\u5bf9\u5e94\u533a\u95f4\u4ea7\u751f\u5f71\u54cd\u3002\u5728\u5b9e\u9645\u751f\u4ea7\u4e2d\uff0c\u7528\u6237\u901a\u5e38\u901a\u8fc7\u52a0\u5165\u8282\u70b9\uff0c\u589e\u52a0\u8282\u70b9\u4e2a\u6570\u7684\u65b9\u5f0f\u83b7\u53d6\u66f4\u591a\u7684\u63a7\u5236\u70b9\uff0c\u8fdb\u800c\u5229\u7528B\u6837\u6761\u7684\u5c40\u90e8\u4fee\u6539\u6a21\u5f0f\u7ec6\u5316\u8c03\u6574\u66f2\u7ebf\uff0c\u800c\u4e0d\u5f71\u54cd\u66f2\u7ebf\u6574\u4f53\u8d70\u52bf\u3002<\/li>\n
      6. \u6837\u6761\u66f2\u7ebf\u5728\u8282\u70b9\u5904\u7684\u8fde\u7eed\u6027\u4e3a\\(C^p\\)\uff0c\u5728\u91cd\u590d\u5ea6\u4e3ak\u7684\u8282\u70b9\u5904\u7684\u8fde\u7eed\u6027\u4e3a\\(C^{p-k}\\)\u3002\u666e\u901a\u8282\u70b9\u7684\u91cd\u590d\u5ea6\u4e3a1\u3002\u56e0\u4e3a\\(C^2\\)\u5df2\u7ecf\u80fd\u591f\u4fdd\u8bc1\u66f2\u7387\u8fde\u7eed\u4e86\uff0c\u6240\u4ee5\u5728\u4e00\u822c\u7684\u4e1a\u52a1\u5e94\u7528\u4e2d\uff0c\u4e09\u6b21(cubic)\u6837\u6761\u66f2\u7ebf\u6700\u4e3a\u5e38\u7528\u3002<\/li>\n
      7. \u53d8\u5dee\u51cf\u5c11\u6027(Variation Diminishing Property)\uff1a\u8fd9\u6761\u6027\u8d28\u4e0eBeizier\u66f2\u7ebf\u7684\u76f8\u8fd1\uff0c\u5982\u679c\u4e00\u6761\u76f4\u7ebf\u4e0e\u6837\u6761\u66f2\u7ebf\u7684\u63a7\u5236\u591a\u8fb9\u5f62\u7684\u4ea4\u70b9\u6570\u4e3aN,\u90a3\u4e48\u8fd9\u6761\u76f4\u7ebf\u4e0e\u6837\u6761\u66f2\u7ebf\u7684\u4ea4\u70b9\u6570\u5e94\u8be5\u4e0d\u5927\u4e8eN\u3002<\/li>\n
      8. Bezier\u66f2\u7ebf\u662f\u7279\u6b8a\u7684\u6837\u6761\u66f2\u7ebf\u3002\u56e0\u4e3aBezier\u66f2\u7ebf\u7684\u63a7\u5236\u70b9\u4e2a\u6570\u4e3a\u6b21\u6570+1\uff0c\u6240\u4ee5\uff0c\u5f53\u5c06\u6837\u6761\u66f2\u7ebf\u7684\u63a7\u5236\u70b9\u53d6p+1\u4e2a\uff0c\u8282\u70b9\u5e8f\u5217\u53d6\\(\\{0,0,…,0,1,1,…1\\}\\)\u66f2\u7ebf\u5373\u4e3abezier\u66f2\u7ebf\u3002<\/li>\n
      9. \u4eff\u5c04\u4e0d\u53d8\u6027\uff1a\u8fd9\u4e00\u6761\u7684\u610f\u601d\u662f\uff0c\u5982\u679c\u9700\u8981\u5bf9\u6837\u6761\u66f2\u7ebf\u505a\u53d8\u6362\uff0c\u90a3\u4e48\u53ea\u9700\u8981\u5bf9\u5176\u63a7\u5236\u70b9\u5e8f\u5217\u505a\u53d8\u6362\u5c31\u53ef\u4ee5\u4e86\u3002\u5176\u53c2\u6570\u533a\u95f4\uff0c\u6b21\u6570\u4fdd\u6301\u4e0d\u53d8\u3002\u975e\u5e38\u597d\u7684\u6027\u8d28\u3002
        \n
        \n

        \u8865\u5145\uff0c\u5f53\\(u=u_i\\)\u65f6\uff0c\\(N_i=0\\)\u3002\u800c\u4e14\u5bf9\u5e94\u533a\u95f4\u4e0a\u81f3\u591a\u6709p+1\u4e2a\u975e\u96f6\u57fa\u51fd\u6570\u3002\u56e0\u6b64\u6027\u8d28\u7b2c4\u6761\u5f3a\u51f8\u5305\u6027\u5e94\u8be5\u662f\u5728\u533a\u95f4\\((u_i,u_{i+1})\\)\u4e14p<i<m-p\u65f6\u6210\u7acb\u3002<\/li>\n<\/ol>\n

        B\u6837\u6761\u7684\u4f18\u70b9\uff1a<\/strong><\/p>\n

        \u5982\u6211\u5728Nurbs\u5386\u53f2<\/a>\u4e2d\u8c08\u5230\u7684\uff0cB\u6837\u6761\u662f\u5bf9\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u4e00\u822c\u5316\u3002\u56e0\u800c\uff0cBezier\u66f2\u7ebf\u80fd\u8868\u8fbe\u7684\uff0cB\u6837\u6761\u4e5f\u540c\u6837\u53ef\u4ee5\u3002\u800c\u4e14\uff0c\u56e0\u4e3aB\u6837\u6761\u5f15\u5165\u4e86\u63a7\u5236\u70b9\u4e2a\u6570\u4e0e\u66f2\u7ebf\u9636\u6b21\u65e0\u5173\u7684\u7279\u6027\uff0c\u4f7f\u5f97\u6211\u4eec\u53ef\u4ee5\u5bf9\u4f4e\u9636\u66f2\u7ebf\u6709\u4e86\u66f4\u597d\u7684\u63a7\u5236\u80fd\u529b\u3002\u800c\u4e14\uff0c\u901a\u8fc7\u5bf9\u8282\u70b9\u5e8f\u5217\u7684\u63a7\u5236\uff0c\u4f7f\u5f97\u6211\u4eec\u5bf9\u66f2\u7ebf\u6709\u81ea\u7531\u7684\u8c03\u8282\u80fd\u529b\u3002”NURBS”\u4ee3\u8868\u7684\u975e\u5747\u5300\u6709\u7406B\u6837\u6761\uff0c\u6211\u4eec\u5c31\u5dee\u201c\u6709\u7406\u201d\u6ca1\u6709\u63a5\u89e6\u4e86\u3002B\u6837\u6761\u4ecd\u7136\u662f\u591a\u9879\u5f0f\u66f2\u7ebf\uff0c\u4e0d\u80fd\u7528\u6765\u8868\u8fbe\u5706\uff0c\u5706\u9525\u66f2\u7ebf\u7b49\u6709\u7406\u66f2\u7ebf\uff0c\u6240\u4ee5NURBS\u662fB\u6837\u6761\u7684\u66f4\u4e00\u822c\u5316\u7684\u5f62\u5f0f\uff0c\u53ef\u4ee5\u8868\u8fbe\u4e0a\u8ff0\u6709\u7406\u66f2\u7ebf\u3002<\/p>\n

        \n

        \u672c\u8282\u53c2\u8003\uff1aIntroduction to Computing with Geometry<\/a>\u00a0 : B-Spline Definition & Important Properties<\/p>\n","protected":false},"excerpt":{"rendered":"

        \u672c\u8282\u7684\u76ee\u6807\u4e3a\u4ecb\u7ecdB\u6837\u6761\u66f2\u7ebf\u7684\u5b9a\u4e49\uff0c\u4ee5\u53ca\u5176\u91cd\u8981\u6027\u8d28\u3002\u8fd9\u4e9b\u6027\u8d28\uff0c\u5bf9\u6837\u6761\u66f2\u7ebf\u7684\u5e94\u7528\u8d77\u4e86 … 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\/493"}],"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=493"}],"version-history":[{"count":8,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/493\/revisions"}],"predecessor-version":[{"id":505,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/493\/revisions\/505"}],"wp:attachment":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=493"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=493"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=493"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}