The calculation proof of the intersection of the space line and the plane and its source code

目的：最近写C++代码,Come across some basic algorithms.The intersection of the space line and the plane is required.

Vector representation of straight lines：$L:\overrightarrow{r}=\overrightarrow{P_0}+t\overrightarrow{N_0}$;其中$\overrightarrow{P_0}=\left(\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right)$, $\overrightarrow{N_0}=\left(\begin{array}{c}{n}_x^0\\ n_y^0\\ n_z^0\end{array}\right)$
Flat vector representation：$\pi :{\overrightarrow{N_1}}^T\cdot X+d_1=0$;其中$\overrightarrow{N_1}=\left(\begin{array}{c}{n}_x^1\\ n_y^1\\ n_z^1\end{array}\right)$,$d$为标量.

For the intersection of the line and the plane, it is divided into three cases：
1）在平面上
2）parallel plane
3）intersects with the plane
图示如下：

It can be seen in the left and middle images,It has no intersection with the plane.Therefore, it is necessary to judge whether the line intersects the plane.in a traditional space,planes and straight planes,The hair vectors of lines and planes are perpendicular.因此可以通过计算$\overrightarrow{N_0}$和 $\overrightarrow{N_1}$之间的夹角.If it is perpendicular to the surface the straight line and the plane are parallel.
$cos(\theta )->{\overrightarrow{N_0}}^T\cdot \overrightarrow{N_1}$
if it's close to0means it is parallel.

The case of intersection is described below
If the plane and the line intersect,对于直线来说,只需要计算$t$即可.
$L:\overrightarrow{r}=\overrightarrow{P_0}+t\overrightarrow{N_0}$带入公式$\pi :{\overrightarrow{N_1}}^T\cdot X+d_1=0$得到如下：
$$\begin{gathered} {\overrightarrow{N_1}}^T\cdot (\overrightarrow{P_0}+t\overrightarrow{N_0})+d_1=0 \\ =>{\overrightarrow{N_1}}^T\cdot \overrightarrow{P_0}+t{\overrightarrow{N_1}}^T\cdot \overrightarrow{N_0}+d_1=0 \\ t=(-d_1-{\overrightarrow{N_1}}^T\cdot \overrightarrow{P_0})/{\overrightarrow{N_1}}^T\cdot \overrightarrow{N_0} \end{gathered}$$
因为向量$\overrightarrow{P_0}$,$\overrightarrow{N_0}$,$\overrightarrow{N_1}$都是已知,So it can be requested$t=t_n$.
带入得到
$$\overrightarrow{r_n}=\overrightarrow{P_0}+t_n\overrightarrow{N_0}$$

Based on the above formula,源码如下：

bool LineRayToPlanePnt(Eigen::Vector3f& o_orign, Eigen::Vector3f& o_dir, Eigen::Vector4f& fn, Eigen::Vector3f& inter_pnt)
	{
    
		Eigen::Vector3f N = Eigen::Vector3f(fn[0], fn[1], fn[2]);
		float D = fn[3];
		if (std::abs(o_dir.dot(N)) < 1e-8)
		{
    
			return false;
		}
		float t = -(o_orign.dot(N) + D) / (o_dir.dot(N));
		inter_pnt = o_orign + t*o_dir;
	}