B样条曲线求解及C++实现 |
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B样条曲线求解及C++实现
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功能：根据参数u值和k（大小为阶数值）与节点矢量，计算第i个k次B样条基数 输入参数： u—参数值；k—大小值为阶数；i—第i个k次B样条的支撑区间左端节点的下标；aNode为节点向量。 输出参数：返回函数值。 double GetBaseFunVal(double u, int i, int k, vector m_aNode) { double Val = 0.0; double Val1 = 0.0; double Val2 = 0.0; if (k==0) { if (u < m_aNode[i] || u > m_aNode[i+1]) return Val; else { Val = 1.0; return Val; } } if (k>0) { if (u < m_aNode[i] || u > m_aNode[i+k+1]) { return Val; } else { double alpha = 0.0; double beta = 0.0; double dTemp = 0.0; dTemp = m_aNode[i+k] - m_aNode[i]; if (dTemp == 0.0) { alpha = 0; } else alpha = (u - m_aNode[i])/dTemp; dTemp = m_aNode[i+k+1] - m_aNode[i+1]; if (dTemp == 0.0) { beta = 0.0; } else beta = (m_aNode[i+k+1] - u)/dTemp; Val1 = alpha * GetBaseFunVal(u, i, k-1, m_aNode); Val2 = beta * GetBaseFunVal(u, i+1, k-1, m_aNode); Val = Val1 + Val2; } } return Val; }上述功能模块摘自于计算机辅助几何设计与非均匀有理B样条。已知B样条的n+1控制点坐标，以及相应的节点向量，可求得对应的曲线方程。 先计算各个控制点的基函数  各个基函数的求解可根据上述的功能模块求出。下面是我的C++实现：曲线是二维的，三维的情况，就Z坐标做同X,Y求解方式相同即可。在求解的过程中，我自己在CAD上画了个样条曲线，然后通过GetBaseFunVal(double u, int i, int k, vector m_aNode)和顶点坐标，及节点向量求各个点的坐标。随着u值的变化，计算各个X,Y,Z值。一个星期的摸爬滚打中，能输出图形，但是与原来的图形对应不上。最终找到的原因在与基函数出问题了。在书本等相关资源中，基函数成员中的k表示的是次数，在我画的样条曲线中，阶数显示为3（为什么是3？CAD的标注里，实体块中的 70 下一行，为3），所以我理所当然的写为了2,。一直有问题。我将它改为3以后，竟然奇迹般的可以用了。而且跟原来图形吻合。这个是我的相关经历，希望对你们能有用。另外，哪位热心人士可以说明下，为什么k改为阶数大小，就可以呢？ // 以下相关数据，坐标点，节点向量，是本人在CAD中绘制一条样条曲线之后得到的数值。关于数据，读者可自行更改。 #include #include #include using namespace std; struct tPoint { double x; double y; double z; }; double GetBaseFunVal(double u, int i, int k, vector m_aNode) { double Val = 0.0; double Val1 = 0.0; double Val2 = 0.0; if (k==0) { if (u < m_aNode[i] || u > m_aNode[i+1]) return Val; else { Val = 1.0; return Val; } } if (k>0) { if (u < m_aNode[i] || u > m_aNode[i+k+1]) { return Val; } else { double alpha = 0.0; double beta = 0.0; double dTemp = 0.0; dTemp = m_aNode[i+k] - m_aNode[i]; if (dTemp == 0.0) { alpha = 0; } else alpha = (u - m_aNode[i])/dTemp; dTemp = m_aNode[i+k+1] - m_aNode[i+1]; if (dTemp == 0.0) { beta = 0.0; } else beta = (m_aNode[i+k+1] - u)/dTemp; Val1 = alpha * GetBaseFunVal(u, i, k-1, m_aNode); Val2 = beta * GetBaseFunVal(u, i+1, k-1, m_aNode); Val = Val1 + Val2; } } return Val; } int main() { tPoint tData; vector vtData; vtData.clear(); vector nodeVector; nodeVector.push_back(0); nodeVector.push_back(0); nodeVector.push_back(0); nodeVector.push_back(0); nodeVector.push_back(1); nodeVector.push_back(2); nodeVector.push_back(3); nodeVector.push_back(4); nodeVector.push_back(5); nodeVector.push_back(6); nodeVector.push_back(6); nodeVector.push_back(6); nodeVector.push_back(6); //节点向量nodeVector， 控制点坐标(0,3),(200,100), (750, 200), k=2 for (double u = 0; u < 6; u=u+0.01) { // 样条的数据 tData.x = (GetBaseFunVal(u, 0, 3, nodeVector)*(-7585) + GetBaseFunVal(u, 1, 3, nodeVector)*(-3427.5) + GetBaseFunVal(u, 2, 3, nodeVector)*46087.5 + GetBaseFunVal(u, 3, 3, nodeVector)*9220.0 + GetBaseFunVal(u, 4, 3, nodeVector)*(-14835.0) + GetBaseFunVal(u, 5, 3, nodeVector)*(-2002.5) + GetBaseFunVal(u, 6, 3, nodeVector)*71975 + GetBaseFunVal(u, 7, 3, nodeVector)*45235 + GetBaseFunVal(u, 8, 3, nodeVector)*83150)/*/(GetBaseFunVal(u, 0, 3, nodeVector) + GetBaseFunVal(u, 1, 3, nodeVector) + GetBaseFunVal(u, 2, 3, nodeVector) + GetBaseFunVal(u, 3, 3, nodeVector) + GetBaseFunVal(u, 4, 3, nodeVector) + GetBaseFunVal(u, 5, 3, nodeVector))*/; tData.y = (GetBaseFunVal(u, 0, 3, nodeVector)*(-3807.5) + GetBaseFunVal(u, 1, 3, nodeVector)*(19850.0) + GetBaseFunVal(u, 2, 3, nodeVector)*14335 + GetBaseFunVal(u, 3, 3, nodeVector)*(-17582.5) + GetBaseFunVal(u, 4, 3, nodeVector)*(-5445.0) + GetBaseFunVal(u, 5, 3, nodeVector)*(-80735.0) + GetBaseFunVal(u, 6, 3, nodeVector)*(-23817.5) + GetBaseFunVal(u, 7, 3, nodeVector)*5037.5 + GetBaseFunVal(u, 8, 3, nodeVector)*(-9360))/*/(GetBaseFunVal(u, 0, 3, nodeVector) + GetBaseFunVal(u, 1, 3, nodeVector) + GetBaseFunVal(u, 2, 3, nodeVector) + GetBaseFunVal(u, 3, 3, nodeVector) + GetBaseFunVal(u, 4, 3, nodeVector) + GetBaseFunVal(u, 5, 3, nodeVector))*/; tData.z = 0.0; vtData.push_back(tData); } char *file = "C:/Users/Monkey/Desktop/新建文件夹 (2)/TEST/Last.txt"; ofstream out(file); if (!out) { cout |
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