{"id":384,"date":"2017-11-30T21:02:37","date_gmt":"2017-11-30T13:02:37","guid":{"rendered":"http:\/\/www.whudj.cn\/?p=384"},"modified":"2017-11-30T21:03:40","modified_gmt":"2017-11-30T13:03:40","slug":"bezier%e6%9b%b2%e7%ba%bf%ef%bc%88%e4%b8%80%ef%bc%89%ef%bc%9a%e5%ae%9a%e4%b9%89%e4%b8%8e%e6%8e%a7%e5%88%b6%e7%82%b9%e7%a7%bb%e5%8a%a8","status":"publish","type":"post","link":"http:\/\/www.whudj.cn\/?p=384","title":{"rendered":"Bezier\u66f2\u7ebf\uff08\u4e00\uff09\uff1a\u5b9a\u4e49\u4e0e\u63a7\u5236\u70b9\u79fb\u52a8"},"content":{"rendered":"

\u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u4e2d\uff0c\u6211\u4eec\u5df2\u7ecf\u63a5\u89e6\u4e86\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u5b9a\u4e49\u3002\u5728\u8fd9\u7bc7\u6587\u7ae0\u91cc\uff0c\u6211\u5c06\u4ece\u51e0\u4f55\u610f\u4e49\u51fa\u53d1\u89e3\u91cabezier\u66f2\u7ebf\u7684\u5b9a\u4e49\u4e0e\u8fd0\u7528\uff0c\u4e3anurbs\u7684\u5b66\u4e60\u5960\u5b9a\u57fa\u7840\u3002<\/p>\n

bezier\u66f2\u7ebf\u7684\u76ee\u7684<\/p>\n

\u8bbe\u8ba1\u4eba\u5458\u5bfb\u627e\u7684\u662f\u66f4\u52a0\u201c\u51e0\u4f55\u201d\u7684\u53c2\u6570\u66f2\u7ebf\uff0c\u5373\u7cfb\u6570\u5177\u6709\u76f4\u63a5\u7684\u51e0\u4f55\u542b\u4e49\uff0c\u80fd\u591f\u901a\u8fc7\u53c2\u6570\u76f4\u63a5\u9884\u6d4b\u51e0\u4f55\u5f62\u72b6\u4ee5\u53ca\u53d8\u5316\u8d8b\u52bf\u7684\u66f2\u7ebf\u3002\u8fd9\u6837\u7684\u66f2\u7ebf\u9700\u8981\u5177\u5907\u4ee5\u4e0b\u7279\u5f81\uff1a<\/p>\n

    \n
  1. \u76f4\u89c2\uff0c\u66f2\u7ebf\u7684\u7b97\u6cd5\u5e94\u5177\u5907\u51e0\u4f55\u610f\u4e49\u3002<\/li>\n
  2. \u7075\u6d3b\uff0c\u66f2\u7ebf\u7684\u521b\u5efa\uff0c\u4fee\u578b\u5e94\u63d0\u4f9b\u66f4\u7075\u6d3b\u7684\u65b9\u5f0f\uff0c\u5e76\u4e14\u521b\u5efa\uff0c\u4fee\u6539\u7684\u65b9\u5f0f\u5e94\u8be5\u66f4\u50cf\u662f\u51e0\u4f55\u64cd\u4f5c\uff0c\u800c\u4e0d\u662f\u4fee\u6539\u65b9\u7a0b\u3002<\/li>\n
  3. \u7edf\u4e00\u65b9\u6cd5\uff0c\u66f2\u7ebf\u80fd\u591f\u8868\u73b0\u5404\u79cd\u57fa\u672c\u5f62\u72b6\u53ca\u5176\u7ec4\u5408\uff0c\u5e76\u4e14\uff0c\u4fee\u6539\u8fd9\u4e9b\u5f62\u72b6\u7684\u65b9\u5f0f\u5e94\u8be5\u662f\u7edf\u4e00\u7684\u3002<\/li>\n
  4. \u9ad8\u6548\u4e14\u5177\u5907\u6570\u503c\u7a33\u5b9a\u6027\u3002<\/li>\n<\/ol>\n

    \u8d1d\u585e\u5c14\u66f2\u7ebf\uff0c\u6837\u6761\u66f2\u7ebf\u3001nurbs\u00a0\u8fd9\u4e00\u7c7b\u66f2\u7ebf\u8f83\u597d\u7684\u89e3\u51b3\u4e86\u4e0a\u8ff0\u95ee\u9898\uff0c\u6210\u4e3a\u5404\u7c7b\u8f85\u52a9\u8bbe\u8ba1\u8f6f\u4ef6(coreldraw\uff0cAutoCAD\u7b49)\u5e7f\u6cdb\u5e94\u7528\u7684\u4e00\u7c7b\u66f2\u7ebf\uff0c\u4ed6\u4eec\u7684\u5171\u540c\u7279\u5f81\u662f\uff1a<\/p>\n

      \n
    1. \u5f15\u5165\u4e86\u201c\u63a7\u5236\u70b9\u201d\u7684\u6982\u5ff5\uff0c\u901a\u8fc7\u63a7\u5236\u70b9\u7684\u5f62\u72b6\u5206\u5e03\uff0c\u80fd\u591f\u786e\u5b9a\u66f2\u7ebf\u7684\u8d70\u52bf\uff0c\u4fee\u6539\u63a7\u5236\u70b9\uff08\u4ee5\u53ca\u5176\u4ed6\u53c2\u6570\uff09\uff0c\u53ef\u4ee5\u5f71\u54cd\u66f2\u7ebf\u7684\u5f62\u72b6\uff0c\u64cd\u4f5c\u76f4\u89c2\uff0c\u4e0d\u9700\u8981\u76f4\u63a5\u4fee\u6539\u65b9\u7a0b\u3002<\/li>\n
    2. \u7528\u6237\u53ef\u4ee5\u4efb\u610f\u7684\u6dfb\u52a0\u63a7\u5236\u70b9\uff0c\u800c\u4e0d\u6539\u53d8\u66f2\u7ebf\u7684\u5f62\u72b6\uff0c\u8fd9\u6837\uff0c\u66f2\u7ebf\u7684\u81ea\u7531\u5ea6\u589e\u52a0\uff0c\u66f2\u7ebf\u4fee\u578b\u5341\u5206\u65b9\u4fbf\u3002<\/li>\n
    3. \u66f2\u7ebf\u5177\u6709\u6253\u65ad\uff0c\u8fde\u63a5\u7684\u80fd\u529b\u3002<\/li>\n
    4. \u66f2\u7ebf\u5230\u66f2\u9762\u7684\u6269\u5c55\u975e\u5e38\u7684\u5bb9\u6613\u3002<\/li>\n<\/ol>\n

      bezier\u66f2\u7ebf\u7684\u5b9a\u4e49\uff1a<\/p>\n

      \u7ed9\u51fa\\(n+1\\)\u4e2a\u63a7\u5236\u70b9\\(\\{P_0,P_1,…,P_n\\}\\)\uff0cn\u9636\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u5b9a\u4e49\u662f \\(C(u)=\\sum_{i=0}^{n}B_{n,i}(u)P_i\\)\u3002\u7cfb\u6570\\(B_{n,i}(u)\\)\u7684\u5b9a\u4e49\u662f\uff1a$$B_{n,i}(u) = \\frac{n!}{i!(n-i)!}u^i(1-u)^{n-i}$$\u8fd8\u8bb0\u5f97\u9ad8\u4e2d\u6570\u5b66\u77e5\u8bc6\u7684\u540c\u5b66\u8981\u7b11\u4e86\uff0c\u563f\u563f\uff0c\u8fd9\u4e2a\u4e0d\u662f\u4e8c\u9879\u5f0f\\((u + (1-u))^n\\)\u7684\u5c55\u5f00\u5f0f\u4e48\uff0c\u771f\u88ab\u4f60\u731c\u5bf9\u4e86\u3002\u5f97\u5230\u8fd9\u4e2a\u89c2\u5bdf\uff0cbezier\u66f2\u7ebf\u7684\u4e00\u4e9b\u91cd\u8981\u7684\u6027\u8d28\u5c31\u7279\u522b\u597d\u7406\u89e3\u4e86\u3002<\/p>\n

      \u7528\u77e9\u9635\u5f62\u5f0f\u8868\u8fbe\u7684bezier\u66f2\u7ebf\u662f\u8fd9\u6837\u7684 $$C(u) = \\begin{bmatrix}B_0&B_1& \\cdots &B_n\\end{bmatrix}\\begin{bmatrix}P_0\\\\P_1\\\\ \\cdots \\\\ P_n\\end{bmatrix}$$\u4e4b\u6240\u4ee5\u8fd9\u4e48\u5199\u662f\u4e3a\u4e86\u8868\u8fbebezier\u662f\u6240\u6709\u63a7\u5236\u70b9\u7ebf\u6027\u7ec4\u5408\u7684\u7ed3\u679c\u3002\u5de6\u4fa7\u7684\u51fd\u6570\u79f0\u4e3a\u57fa\u51fd\u6570\uff0c\u53f3\u4fa7\u7684\u70b9\u79f0\u4e3a\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u201c\u63a7\u5236\u70b9\u201d\u3002\u53c2\u6570u\u7684\u5b9a\u4e49\u57df\u4e3a[0,1],\u56e0\u6b64\uff0c\u6240\u6709\u7684\u57fa\u51fd\u6570\u90fd\u662f\u975e\u8d1f\u7684\u3002\u56e0\u4e3au\u4e0ei\u90fd\u53ef\u80fd\u4e3a\u96f6\uff0c\u6211\u4eec\u89c4\u5b9a\\(0_0=1\\)\u3002<\/p>\n

      \u8d1d\u585e\u5c14\u66f2\u7ebf\u6709\u5982\u4e0b\u91cd\u8981\u7684\u6027\u8d28\u3002<\/p>\n

        \n
      1. n+1\u4e2a\u63a7\u5236\u70b9\u7684\u5ea6\u6570\u4e3an\u3002<\/li>\n
      2. \u66f2\u7ebf\u901a\u8fc7\\(P_0\\)\u4e0e\\(P_n\\)\u3002\u5f53u=0\u65f6\uff0c\u66f2\u7ebf\u901a\u8fc7\\(P_0\\),\u5f53u=1\u65f6\uff0c\u66f2\u7ebf\u901a\u8fc7\\(P_n\\)\uff0c\u8fd9\u6bd4\u8f83\u65b9\u4fbf\u505a\u5f62\u72b6\u7684\u63a7\u5236\u3002<\/li>\n
      3. \u975e\u8d1f\u6027\uff0c\u6240\u6709\u7684\u57fa\u51fd\u6570\u90fd\u662f\u975e\u8d1f\u7684\u3002<\/li>\n
      4. \u5f52\u4e00\u6027\uff0c\u5728\u67d0\u4e2a\u7279\u5b9a\u7684\u53c2\u6570u\u5904\uff0c\u6240\u6709\u7684\u57fa\u51fd\u6570\u7684\u548c\u52a0\u8d77\u6765\u90fd\u7b49\u4e0e1\u3002\u524d\u9762\u6211\u4eec\u8bf4\u8fc7\u00a0\u57fa\u51fd\u6570\u5176\u5b9e\u662f\u4e8c\u9879\u5f0f\\((1+(1-u))^n=1^n\\)\u7684\u5c55\u5f00\u5f0f\uff0c\u5176\u52a0\u548c\u80af\u5b9a\u7b49\u4e0e1\u3002\u4e0b\u56fe\u662f\u56db\u9636bezier\u66f2\u7ebf\u7684\u57fa\u51fd\u6570\u56fe\u50cf\u3002<\/li>\n
      5. \"\"<\/li>\n
      6. \u00a0\u51f8\u5305\u6027\u8d28\uff0c\u66f2\u7ebf\u5904\u5728\u6240\u6709\u63a7\u5236\u70b9\u6240\u51b3\u5b9a\u7684\u51f8\u5305\u4e4b\u5185\uff0c\u5e76\u4e14\u53ea\u6709\u9996\u5c3e\u70b9\u4f4d\u4e8e\u51f8\u5305\u4e4b\u4e0a\u3002\u8fd9\u4e2a\u6027\u8d28\u5728bezier\u8f6c\u6298\u7ebf\uff0cbezier\u7a7a\u95f4\u8303\u56f4\u5224\u65ad\u7b49\u8ba1\u7b97\u4e0a\u975e\u5e38\u6709\u7528\u3002<\/li>\n
      7. Variation Diminishing Property\uff1a\u8fd9\u4e00\u6761\u771f\u5fc3\u4e0d\u77e5\u9053\u600e\u4e48\u7ffb\u8bd1\uff0c\u610f\u601d\u5c31\u662f\u4e00\u6761\u76f4\u7ebf\u4e0ebezier\u7684\u4ea4\u70b9\u6570\u91cf\u4e00\u5b9a\u4e0d\u5927\u4e8e\u76f4\u7ebf\u4e0e\u66f2\u7ebf\u63a7\u5236\u591a\u8fb9\u5f62\u4ea4\u70b9\u7684\u6570\u91cf\u3002\u5982\u679c\u66f2\u7ebf\u4e3a\u7a7a\u95f4\u66f2\u7ebf\uff0c\u5c06\u7ebf\u66ff\u6362\u6210\u5e73\u9762\u540c\u6837\u53ef\u4ee5\u6ee1\u8db3\u4e0a\u8ff0\u6027\u8d28\u3002\u8fd9\u6761\u6027\u8d28\u8bf4\u660e\u7684\u662fbezier\u66f2\u7ebf\u7684\u626d\u8f6c\u548c\u5f2f\u6298\u7684\u60c5\u51b5\u8981\u6bd4\u5176\u63a7\u5236\u591a\u8fb9\u5f62\u8981\u8f7b\u5fae\u3002<\/li>\n
      8. \u4eff\u5c04\u4e0d\u53d8\u6027\uff1a\u5982\u679c\u9700\u8981\u5bf9bezier\u66f2\u7ebf\u505a\u4eff\u5c04\u53d8\u6362\uff0c\u53ea\u9700\u8981\u5bf9\u5176\u63a7\u5236\u70b9\u5e8f\u5217\u8fdb\u884c\u4eff\u5c04\u53d8\u6362\u5373\u53ef\u3002\u5176\u57fa\u51fd\u6570\uff08\u77e9\u9635\u5f62\u5f0f\u7684\u5de6\u534a\u90e8\u5206\uff09\u4e0d\u9700\u8981\u53d8\u5316\u3002\u8fd9\u4e2a\u6027\u8d28\u4f7f\u5f97bezier\u66f2\u7ebf\u5bf9\u4e8e\u56fe\u5f62\u53d8\u5316\u975e\u5e38\u53cb\u597d\u3002<\/li>\n<\/ol>\n

        \u53c2\u6570\u5b9a\u4e49\u57df\u4e0d\u5728[0,1]\u7684\u8d1d\u585e\u5c14\u66f2\u7ebf<\/p>\n

        \u5bf9\u4e8e\u5b9a\u4e49\u57df\u4e0d\u5728[0,1]\u5185\u7684\u8d1d\u585e\u5c14\u66f2\u7ebf\uff0c\u53ef\u4ee5\u901a\u8fc7\u62c9\u4f38\u7684\u65b9\u5f0f\u5c06\u5176\u6620\u5c04\u5230[0,1]\uff0c\u5047\u8bbe\u5b9a\u4e49\u57df\u4e3a[a,b]\u7684\u66f2\u7ebf\uff0c\u5176\u57fa\u51fd\u6570\u7684\u5f62\u5f0f\u4e3a$$B_{n,i}(u) = \\frac{n!}{i!(n-i)!(b-a)^n}(u-a)^i(b-u)^{n-i}$$<\/p>\n

        \u79fb\u52a8\u63a7\u5236\u70b9<\/p>\n

        \u5982\u679c\u79fb\u52a8\u67d0\u63a7\u5236\u70b9\\(P_k\\)\u5230\u65b0\u4f4d\u7f6e\\(P_k+\\vec v\\),\u90a3\u4e48bezier\u66f2\u7ebf\u7684\u5f62\u72b6\u5c06\u53d1\u751f\u4ec0\u4e48\u6837\u7684\u53d8\u5316\u5462\uff1f\u5047\u8bbe\u65b0\u7684bezier\u66f2\u7ebf\u662f\\(D(u)\\)\uff0c$$D(u) = \\begin{bmatrix}B_0& \\cdots &B_k& \\cdots &B_n\\end{bmatrix}\\begin{bmatrix}P_0\\\\\\cdots \\\\P_k+\\vec v\\\\ \\cdots \\\\ P_n\\end{bmatrix} = C(u) + B_k\\vec v$$\u5373\u65b0\u66f2\u7ebf\u7684\u6bcf\u4e2a\u70b9\u7684\u79fb\u52a8\u8ddd\u79bb\u4e3a\u5176\u5bf9\u5e94\u7684\u53c2\u6570u\u7684\u57fa\u51fd\u6570\\(B_k(u)\\)\u4e58\u4ee5\u79fb\u52a8\u5411\u91cf\\(\\vec v\\)\u3002\u66f2\u7ebf\u79fb\u52a8\u7684\u65b9\u5411\u90fd\u662f\u4e00\u81f4\u7684(\u57fa\u51fd\u6570\u975e\u8d1f)\uff0c\u5728\\(B_k(u)\\)\u8fbe\u5230\u6700\u5927\u503c\u65f6\uff0c\u79fb\u52a8\u7684\u8ddd\u79bb\u8fbe\u5230\u6700\u5927\u503c\u3002\u800c\u4e14\u56e0\u4e3a\u57fa\u51fd\u6570\u5c0f\u4e8e\u7b49\u4e8e1\uff0c\u66f2\u7ebf\u4e0a\u6bcf\u4e2a\u70b9\u7684\u6a21\u957f\u5747\u5c0f\u4e8e\u7b49\u4e8e\\(|\\vec v|\\)\u3002<\/p>\n

        \n

        \u672c\u6587\u53c2\u8003Introduction to Computing with Geometry \u7684\u7b2c\u4e94\u7ae0 Bezier Curves \u7684\u524d\u4e09\u8282\u3002<\/p>\n","protected":false},"excerpt":{"rendered":"

        \u8ba1\u7b97\u673a\u56fe\u5f62\u5b66\u4e2d\uff0c\u6211\u4eec\u5df2\u7ecf\u63a5\u89e6\u4e86\u8d1d\u585e\u5c14\u66f2\u7ebf\u7684\u5b9a\u4e49\u3002\u5728\u8fd9\u7bc7\u6587\u7ae0\u91cc\uff0c\u6211\u5c06\u4ece\u51e0\u4f55\u610f\u4e49\u51fa\u53d1 … 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\/384"}],"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=384"}],"version-history":[{"count":18,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/384\/revisions"}],"predecessor-version":[{"id":406,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=\/wp\/v2\/posts\/384\/revisions\/406"}],"wp:attachment":[{"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=384"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Fcategories&post=384"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.whudj.cn\/index.php?rest_route=%2Fwp%2Fv2%2Ftags&post=384"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}