eryar@163.com

Abstract. The natural parametric equations of a curve are parametric equations that represent the curve in terms of a coordinate-independent parameter, generally arc length s, instead of an arbitray variable like t or u. According to the natural equations, the curve length is the integration of the curve parametric equation’s derivation. So the core algorithm for curve length calculation is the numerical integration method. OpenCASCADE use Gauss-Legendre to calculate the integration for single variable and multiple variables. Because of curve in OpenCASCADE is single variable function, so can use the Gauss-Legendre to calculate the arc length for the curve.

Key Words. OpenCASCADE, The Gauss-Legendre Integration, Parametric Curve,

The Natural Parametric Equations,  Arc Length,

1. Introduction

Figure 1.1 Length of Curve

2. The Natural Parametric Equations

Figure 2.1 Arc length parameterization

s(u)即为与参数u0和u1对应的两点P0和P之间的弧长。弧长参数化是一类重要的概念，但是如果用上式来计算弧长比较繁琐，可以通过累加弦长方法来近似计算。

Figure 2.2 Parametric Curve

3. The Gauss-Legendre Integration

Figure 3.1 Gauss Point for Gauss-Legendre Integration

Gauss-Legendre积分用处之一就是根据曲线的自然参数方程计算曲线的弧长，代码实现如下所示：

```//=====================================================================
//function : Length
//purpose  : 3d with parameters
//=====================================================================

const Standard_Real U1,
const Standard_Real U2)
{
CPnts_MyGaussFunction FG;
//POP pout WNT
CPnts_RealFunction rf = f3d;
math_GaussSingleIntegration TheLength(FG, U1, U2, order(C));
if (!TheLength.IsDone()) {
Standard_ConstructionError::Raise();
}
return Abs(TheLength.Value());
}```

4. Conclusion

5. References

1. 朱心雄. 自由曲线曲面造型技术. 科学出版社. 2008

2. 王仁宏, 李崇君, 朱春钢. 计算几何教程. 科学出版社. 2008

3. 易大义, 沈云宝, 李有法. 计算方法. 浙江大学出版社. 2002

4. 易大义, 陈道琦. 数值分析引论. 浙江大学出版社. 1998