{"id":585,"date":"2017-12-17T22:26:00","date_gmt":"2017-12-17T14:26:00","guid":{"rendered":"http:\/\/www.whudj.cn\/?p=585"},"modified":"2017-12-19T18:58:06","modified_gmt":"2017-12-19T10:58:06","slug":"b-spline%e4%b9%9d%e6%89%93%e6%96%adsubdividing%e4%bb%a5%e5%8f%8a%e6%8a%98%e7%ba%bf%e5%8c%96tesslelation","status":"publish","type":"post","link":"http:\/\/www.whudj.cn\/?p=585","title":{"rendered":"B-Spline(\u4e5d):\u6253\u65ad(Subdividing)\u4ee5\u53ca\u6298\u7ebf\u5316(Tesslelation)"},"content":{"rendered":"

\u4e0eBezier\u66f2\u7ebf\u7684\u6253\u65ad<\/a>\u65b9\u6cd5\u7c7b\u4f3c\uff0cB\u6837\u6761\u7684\u6253\u65ad\u5229\u7528\u4e86de Boor\u7b97\u6cd5<\/a>\u3002\u5e76\u4e14\u7ed3\u5408B\u6837\u6761\u7684\u5f3a\u51f8\u5305\u6027<\/a>\uff0c\u6211\u4eec\u53ef\u4ee5\u63a8\u7b97\u51fa\u4e00\u79cd\u6709\u6548\u7684B\u6837\u6761\u6298\u7ebf\u5316\u65b9\u6cd5\u3002\u56e0\u6b64\uff0c\u672c\u8282\u662f\u5bf9\u524d\u9762\u51e0\u8282\u5185\u5bb9\u7684\u4e00\u4e2a\u7efc\u5408\u5e94\u7528\uff0c\u540e\u9762\uff0c\u6211\u7684\u5206\u4eab\u4e5f\u4e3b\u8981\u9762\u5411\u5e94\u7528\u3002<\/p>\n

1\uff09B\u6837\u6761\u7684\u6253\u65ad<\/strong><\/p>\n

\u7ed9\u5b9a\u53c2\u6570u,\u5e0c\u671b\u5728u\u5904\u5c06b\u6837\u6761\u5206\u4e3a\u4e24\u90e8\u5206\u3002\u8fd9\u4e2a\u8fc7\u7a0b\u4e0eBezier\u66f2\u7ebf\u7684\u6253\u65ad\u975e\u5e38\u76f8\u4f3c\u3002\u53ea\u4e0d\u8fc7\uff0cBezier\u7684\u6253\u65ad\u662f\u5728\u6240\u6709\u7684\u63a7\u5236\u70b9\u4e0a\u8fdb\u884c\u201c\u5207\u89d2\u201d\uff0c\u800c\u5bf9\u4e8e\\(u\\in[u_k,u_{k+1})\\)\uff0c\u4ec5\\(P_{k-p},…,P_k\\)p+1\u4e2a\u63a7\u5236\u70b9\u88ab\u5207\u89d2(\u5176\u5b9eBezier\u66f2\u7ebf\u4e5f\u662fp+1\u4e2a\u63a7\u5236\u70b9\u2026)\u3002\u4e0b\u56fe\u4ee5\u4e09\u9636B\u6837\u6761\u4e3a\u4f8b\uff0c\u5c55\u793a\u6253\u65ad\u8fc7\u7a0b\uff1a<\/p>\n

\"\"<\/p>\n

\u6211\u5206\u522b\u7528Q,R,S\u4ee3\u8868\u4e09\u6b21\u201c\u5207\u89d2\u201d\u3002de Boor\u7b97\u6cd5\u6700\u540e\u8ba1\u7b97\u51fa\u7684C(u)\u5373\\(S_k\\)\u3002\u7eff\u8272\u7684\u70b9\\(P_{k-3},Q_{k-2},R_{k-1},S_{k}\\)\u4e3a\u6253\u65ad\u540e\u5de6\u8fb9\u7684B\u6837\u6761\u7684\u65b0\u63a7\u5236\u70b9\uff0c\\(S_{k},R_{k},Q_{k},P_{k}\\)\u4e3a\u6253\u65ad\u540e\u53f3\u4fa7\u7684B\u6837\u6761\u7684\u65b0\u63a7\u5236\u70b9\u3002<\/p>\n

\u6211\u7684C++\u7248\u672c\u7684\u6253\u65ad\u5b9e\u73b0\u5982\u4e0b\uff1a<\/p>\n

void BSpline::Subdeviding(double u,BSpline& sub_left,BSpline& sub_right)\r\n{\r\n\t\/\/\u4e3a\u4e86\u4f7f\u7528\u65b9\u4fbf\r\n\tconst std::vector<double>& knots = m_vecKnots;\r\n\tconst std::vector<Point>& cv = m_vecCVs;\r\n\tint p = m_nDegree;\r\n\r\n\t\/\/\u67e5\u627e u \u6240\u5728\u533a\u95f4\r\n\tint k = std::distance(knots.begin(),std::upper_bound(knots.begin(), knots.end(), u))-1;\r\n\r\n\t\/\/\u4fdd\u5b58de boor \u7b97\u6cd5\u8fed\u4ee3\u8fc7\u7a0b\u4e2d\u751f\u6210\u7684\u63a7\u5236\u70b9\r\n\tstd::vector<Point> cv_left,cv_right;\r\n\t_subdeviding(u,k,knots,cv,cv_left,cv_right);\r\n\r\n\t\/\/\u66f2\u7ebf\u5728u\u5904\u6253\u65ad\uff0c\u751f\u6210\u65b0\u7684\u63a7\u5236\u70b9\u5e8f\u5217 \u5047\u8bbe\u539f\u66f2\u7ebf\u7684\u63a7\u5236\u70b9\u5e8f\u5217\u4e3a P_0,...,P_k-p-1,P_k-p,...,P_k,P_k+1,...,P_n\r\n\t\/\/\u6253\u65ad\u540e\u7684\u65b0\u63a7\u5236\u70b9\u5e8f\u5217\u4e3a\uff1a\u5de6\u4fa7\uff1aP_0,....,P_k-p-1,cv_left[...]\uff0c\u53f3\u4fa7 cv_right[...],P_k+1,...,P_n\u3002\r\n\t\/\/sub_left\r\n\tsub_left.m_nDegree = p;\r\n\t\/\/knots\r\n\tsub_left.m_vecKnots.resize(k+p+2);\r\n\tfor (int i=0;i<k+1;++i)\r\n\t\tsub_left.m_vecKnots[i] = knots[i];\r\n\tfor (int i=0,j=k+1;i<p+1;++i,++j)\r\n\t\tsub_left.m_vecKnots[j] = u;\r\n\t\/\/control vertex\r\n\tsub_left.m_vecCVs.resize(k+1);\r\n\tfor (int i=0;i<k-p;++i)\r\n\t\tsub_left.m_vecCVs[i] = cv[i];\r\n\tfor (int i=0,j=k-p;i<p+1;++i,++j)\r\n\t\tsub_left.m_vecCVs[j] = cv_left[i];\r\n\t\r\n\t\/\/sub_right\r\n\tsub_right.m_nDegree = p;\r\n\t\/\/knots\r\n\tsub_right.m_vecKnots.resize(knots.size()-k+p);\r\n\tfor (int i=0;i<p+1;i++)\r\n\t\tsub_right.m_vecKnots[i] =u;\r\n\tfor (int i=p+1,j=k+1;j<knots.size();++j,++i)\r\n\t\tsub_right.m_vecKnots[i]=knots[j];\r\n\t\/\/control_vertex\r\n\tsub_right.m_vecCVs.resize(cv.size()-k+p);\r\n\tfor (int i=0;i<p+1;++i)\r\n\t\tsub_right.m_vecCVs[i] = cv_right[i];\r\n\tfor (int i=k+1,j=p+1;i<cv.size();++i,++j)\r\n\t\tsub_right.m_vecCVs[j] = cv[i]; \r\n}\r\n\r\nvoid BSpline::_subdeviding(double u,int k,const std::vector<double>& knots,const std::vector<Point>& cv, \r\n\t\t\t\t\t\t   std::vector<Point>& cv_left,std::vector<Point>& cv_right)\r\n{\r\n\tint p= m_nDegree;\r\n\r\n\t\/\/\u4fdd\u5b58de boor \u7b97\u6cd5\u8fed\u4ee3\u8fc7\u7a0b\u4e2d\u751f\u6210\u7684\u63a7\u5236\u70b9\r\n\tcv_left.resize(p+1);\r\n\tcv_right.resize(p+1);\r\n\r\n\t\/\/\u5c06 P_k-p,...,P_k\u62f7\u8d1d\u5230cv_left\u4e0a\u9762\r\n\tstd::copy(cv.begin() + k - p, cv.begin() + k + 1, cv_left.begin());\r\n\tcv_right[p] = cv_left[p];\r\n\r\n\t\/\/de-boor \u8fed\u4ee3p\u6b21\r\n\tfor (int r = 1; r <= p; ++r)\r\n\t{\r\n\t\t\/\/i \u4ece k \u5230 k-p+1\r\n\t\tfor (int i = k,j=p; i >= k - p + r; --i,--j)\r\n\t\t{\r\n\t\t\tdouble alpha = u - knots[i];\r\n\t\t\tdouble dev =  (knots[i + p + 1 - r] - knots[i]);\r\n\t\t\talpha = (dev !=0)?  alpha\/dev : 0;\r\n\r\n\t\t\tcv_left[j] = (1.0 - alpha)*cv_left[j - 1] + alpha * cv_left[j];\r\n\t\t}\r\n\r\n\t\tcv_right[p-r] = cv_left[p];\r\n\t}\r\n}<\/pre>\n
\u4e0b\u56fe\u662f\u6253\u65ad\u7b97\u6cd5\u5bf9\u67d0\u4e00\u6837\u6761\u5728\\(u=3000\\)\u5904\u6253\u65ad\u7684\u6548\u679c\u3002\u4e3a\u4e86\u65b9\u4fbf\u5c55\u793a\uff0c\u4e24\u6bb5\u6253\u65ad\u540e\u7684\u6837\u6761\u66f2\u7ebf\u90fd\u7ecf\u8fc7\u4e86\u79fb\u52a8\u3002<\/p>\n
\"\"<\/p>\n
2<\/strong>\uff09B\u6837\u6761\u7684\u6298\u7ebf\u5316<\/strong><\/p>\n
B\u6837\u6761\u7684\u6298\u7ebf\u5316\u4e5f\u662f\u6700\u5e38\u7528\u7684\u529f\u80fd\u4e4b\u4e00\u3002\u6bd4\u5982\u66f2\u7ebf\u8f6c\u6362\u6210\u6298\u7ebf\uff0c\u66f2\u7ebf\u6e32\u67d3\u7b49\u7b49\u90fd\u9700\u8981\u5c06\u7528\u4e00\u7cfb\u5217\u79bb\u6563\u70b9\u4ee3\u8868B\u6837\u6761\u66f2\u7ebf\u3002\u56e0\u6b64\u4e5f\u53ef\u4ee5\u628a\u8fd9\u4e2a\u8fc7\u7a0b\u79f0\u4e4b\u4e3a\u66f2\u7ebf\u7684\u7ec6\u5206(Tesselation)\u3002\u4e00\u79cd\u7b80\u5355\u7684(naive)\u60f3\u6cd5\u662f\u5747\u5300\u7ec6\u5206\uff0c\u5373\u5c06\u8282\u70b9\u533a\u95f4\u5747\u5300\u7684\u5212\u5206\u4e3ak\u4efd\uff0c\u7136\u540e\u5bf9\u65b0\u8282\u70b9\u5206\u522b\u6c42\u503c\uff0c\u7b97\u51fa\u66f2\u7ebf\u4e0a\u7684\u70b9\u4ee3\u66ff\u539f\u66f2\u7ebf\u3002\u5982\u4e0b\u56fe\u793a\uff1a<\/p>\n
\"\"<\/p>\n
\u8fd9\u79cd\u5212\u5206\u6bd4\u8f83\u7b80\u5355\uff0c\u4f46\u662f\u7f3a\u70b9\u4e5f\u6bd4\u8f83\u660e\u663e\uff0c\u5373\u53c2\u6570\u7684\u6b65\u957f\u4e0d\u597d\u786e\u5b9a\u3002\u6b65\u957f\u8fc7\u5927\u5bf9\u4e8e\u5f2f\u66f2\u7684\u66f2\u7ebf\u5bb9\u6613\u9020\u6210\u6298\u7ebf\u6298\u89d2\u8fc7\u5927\uff0c\u6b65\u957f\u8fc7\u5c0f\u4f1a\u9020\u6210\u6570\u636e\u5197\u4f59\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u9700\u8981\u4e00\u4e2a\u81ea\u9002\u5e94\u7684\u65b9\u6cd5\u3002\u56de\u60f3B\u6837\u6761\u7684\u4e00\u4e2a\u6027\u8d28\u662f\u201c\u5f3a\u51f8\u5305\u6027\u201d\uff0c\u8bf4\u7684\u662fB\u6837\u6761\u66f2\u7ebf\u4e25\u683c\u5305\u542b\u5728\u63a7\u5236\u70b9\u6784\u6210\u7684\u51f8\u5305\u4e4b\u5185\u3002\u56e0\u6b64\uff0c\u6211\u4eec\u53ef\u4ee5\u901a\u8fc7\u51f8\u5305\u7684\u5f62\u72b6\u9884\u6d4b\u66f2\u7ebf\u7684\u5f62\u72b6\u3002\u5f53\u51f8\u5305\u5f62\u72b6\u903c\u8fd1\u76f4\u7ebf\u65f6\uff0c\u66f2\u7ebf\u7684\u5f62\u72b6\u4e5f\u903c\u8fd1\u76f4\u7ebf\u3002\u81ea\u9002\u5e94\u7b97\u6cd5\u91c7\u7528\u201c\u5206\u800c\u6cbb\u4e4b\u201d\u7684\u601d\u8def\uff0c\u9012\u5f52\u7684\u5c06\u66f2\u7ebf\u5206\u6210\u4e24\u90e8\u5206\u505a\u7ec6\u5206\u8fd0\u7b97\uff1a<\/p>\n
    \n
  1. \u7ed9\u5b9a\u8282\u70b9\u9996\u5c3e \\(u_{beg},u_{end}\\),\u8ba1\u7b97\u6240\u6709\u63a7\u5236\u70b9\u5230\u9996\u3001\u5c3e\u63a7\u5236\u70b9\u7684\u6700\u9ad8\u9ad8\u5ea6\uff0c\u5982\u679c\u6700\u9ad8\u9ad8\u5ea6\u5c0f\u4e8e\u9608\u503cd\uff0c\u90a3\u4e48\u66f2\u7ebf\u8db3\u591f\u76f4\uff0c\u8ba1\u7b97\u7ec8\u6b62\uff0c\u8f93\u51fa\u6298\u7ebf\u5316\u7ed3\u679c\u3002<\/li>\n
  2. \u5982\u679c\u4e0d\u6ee1\u8db3\u9608\u503c\uff0c\u5c06\u66f2\u7ebf\u5728\u00a0\\(u_{mid}=(u_{beg}+u_{end})\/2\\)\u5904\u6253\u65ad\uff0c\u5206\u6210\u4e24\u90e8\u5206\uff0c\u5e76\u4e14\u5206\u522b\u5c06\u4e24\u4e2a\u90e8\u5206\u6267\u884c\u6b65\u9aa41\u3002<\/li>\n<\/ol>\n
    \u6211\u7684\u7b97\u6cd5\u7684C++\u7684\u5b9e\u73b0\u662f\uff1a<\/p>\n
    void BSpline::Tesselation(double tolerance ,std::vector<Point>& points)\r\n{\r\n\tstd::vector<Point> cv_copy = m_vecCVs;\r\n\tstd::vector<double> knots_copy = m_vecKnots;\r\n\t_tesselation(cv_copy, knots_copy, points, tolerance, m_nDegree, m_vecKnots.size() - m_nDegree,0,m_vecCVs.size()-1);\r\n}\r\n\r\nvoid BSpline::_tesselation(std::vector<Point>& cv, std::vector<double>& knots, std::vector<Point>& points, double tolerance, int k_s, int k_e,int c_s,int c_e)\r\n{\r\n\tint p = m_nDegree;\r\n\r\n\tbool stright_enough = true;\r\n\t\/\/\u8ba1\u7b97\u5f26\u9ad8\u662f\u5426\u8d85\u8fc7\u5bb9\u5dee\r\n\tfor (int i = c_s + 1; i < c_e; ++i)\r\n\t{\r\n\t\t\/\/\u70b9\u5230\u76f4\u7ebf\u8ddd\u79bb\u516c\u5f0f\uff0c\u4e0d\u7ed9\u51fa\u5b9e\u73b0\u4e86\r\n\t\tif (PointLineDistance(cv[i], cv[c_s], cv[c_e]) > tolerance)\r\n\t\t{\r\n\t\t\tstright_enough = false;\r\n\t\t\tbreak;\r\n\t\t}\r\n\t}\r\n\r\n\t\/\/\u6ee1\u8db3\u8981\u6c42\uff0c\u4e0d\u8fdb\u4e00\u6b65\u7ec6\u5206\u4e86\r\n\tif (stright_enough)\r\n\t{\r\n\t\t\/\/\u4e3a\u4e86\u4fdd\u8bc1\u63a7\u5236\u70b9\u4e0d\u91cd\u590d\uff0c\u8bbe\u8ba1\u7684\u89c4\u5219\u4e3a[),\u4f46\u662f\u5bf9\u6700\u540e\u4e00\u4e2a\u70b9\u4f8b\u5916\u3002\r\n\t\t\/\/\u6309\u7167\u9012\u5f52\u987a\u5e8f\uff0c\u6700\u540e\u4e00\u6bb5\u9996\u5148\u52a0\u5165points\r\n\t\tint c_end = points.empty() ? c_e +1: c_e;\r\n\t\tpoints.insert(points.begin(),cv.begin()+c_s,cv.begin()+c_end);\r\n\t\treturn;\r\n\t}\r\n\r\n\t\/\/\u4ece\u8282\u70b9\u4e2d\u95f4\u6253\u65ad\r\n\tdouble u_mid = knots[k_s] + (knots[k_e] - knots[k_s]) \/ 2.0;\r\n\t\/\/\u67e5\u627e u \u6240\u5728\u533a\u95f4\r\n\tint k = std::distance( knots.begin(),std::upper_bound(knots.begin(), knots.end(), u_mid)) - 1;\r\n\r\n\tstd::vector<Point> cv_left, cv_right;\r\n\t_subdeviding(u_mid,k,knots,cv,cv_left,cv_right);\r\n\t\r\n\t\/\/\u8282\u70b9\u533a\u95f4\u65b0\u589ep\u4e2au_mid\r\n\tknots.insert(knots.begin() + k+1, p,u_mid);\r\n\t\/\/\u63a7\u5236\u70b9\u66ff\u6362\r\n\tcv.insert(cv.begin() + k, p,Point());\r\n\tfor (int i = k - p, j = 0; j <p; ++j, ++i)\r\n\t\tcv[i] = cv_left[j];\r\n\r\n\tfor (int i = k, j = 0; j <= p; ++j, ++i)\r\n\t\tcv[i] = cv_right[j];\r\n\r\n\t\/\/\u4e24\u90e8\u5206\u5206\u522b\u9012\u5f52\r\n\t\/\/Note:\u540e\u534a\u90e8\u5206\u5728\u524d\u534a\u90e8\u5206\u4e4b\u524d\u6267\u884c\uff0c\r\n\t\/\/\u56e0\u4e3a\u5982\u679c\u524d\u534a\u90e8\u5206\u9996\u5148\u6267\u884c\u7684\u5316\uff0c\u540e\u534a\u90e8\u5206\u7684\u7d22\u5f15\u5c31\u53d1\u751f\u6539\u53d8\u4e86\r\n\t_tesselation(cv, knots, points, tolerance, k + 1, k_e + p,k,c_e+p);\r\n\t_tesselation(cv, knots, points, tolerance, k_s, k + 1, c_s, k);\r\n}<\/pre>\n
    \u4e0b\u56fe\u662f\u5bf9\u67d0\u67613\u9636B\u6837\u6761\u66f2\u7ebf\u8fdb\u884ctorlerance =1 \u7684\u6298\u7ebf\u5316\u5c40\u90e8\u6548\u679c\uff0c\u53ef\u4ee5\u770b\u5230\uff1a\u6298\u7ebf\u5728\u5f2f\u66f2\u7684\u5730\u65b9\u70b9\u6bd4\u8f83\u5bc6\uff0c\u800c\u5e73\u76f4\u7684\u5730\u65b9\u70b9\u6bd4\u8f83\u5c11\uff0c\u6240\u4ee5\u5177\u5907\u8f83\u597d\u7684\u81ea\u9002\u5e94\u7684\u7279\u70b9\u3002<\/p>\n
    \"\"<\/p>\n
    3\uff09\u6298\u7ebf\u5316\u7b97\u6cd5\u5206\u6790\uff1a<\/strong><\/p>\n
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