二维螺旋曲线方程式，弧长计算及作图实现qq23516957的博客-

设R1为内径，n为圈数，d为圈距。

• 参数方程式如下：

• 极坐标方程如下：

• 计算长度可近似为：

• 举例证明

如果内半径为R1 = 5，并且每转弯处半径增加为d=0.81，圈数n = 7.5。

利用计算得：

而使用极坐标中曲线的弧长的公式来计算计算得：

两者近似相等。

• qt作图如下

• 源码

   // create empty curve objects:     QCPCurve *fermatSpiral1 = new QCPCurve(m_customPlot->xAxis, m_customPlot->yAxis);     // set the same step between xAxis and yAxis     QCPAxisTickerFixed *ticker = new QCPAxisTickerFixed;     ticker->setTickStep(1);     m_customPlot->xAxis->setTicker(QSharedPointer<QCPAxisTicker>(ticker));     m_customPlot->yAxis->setTicker(QSharedPointer<QCPAxisTicker>(ticker));     // generate the curve data points:     const int pointCount = 500;     QVector<QCPCurveData> dataSpiral1(pointCount);     for (int i=0; i<pointCount; ++i)     {         double phi = i/(double)(pointCount-1);         double N = 7.5;//圈数         double d_Inner = 5;//内径         double d = 0.81;//圈距         double Dis =N*d*phi;//外径 - 内径         dataSpiral1[i] = QCPCurveData(i, (d_Inner + Dis)*qCos(N*phi*2*M_PI), (d_Inner + Dis)*qSin(N*phi*2*M_PI));     }     // pass the data to the curves; we know t (i in loop above) is ascending, so set alreadySorted=true (saves an extra internal sort):     fermatSpiral1->data()->set(dataSpiral1, true);     // color the curves:     fermatSpiral1->setPen(QPen(Qt::blue));