Calculation Proof and Source Code of Intersection Points from Space Lines to Planes

Purpose: I recently wrote C++ code and encountered some basic algorithms. The intersection of a line and a plane in space is required.

Vector representation of a straight line: $L:\overrightarrow{r}=\overrightarrow{P_0}+t\overrightarrow{N_0}$；其中$\overrightarrow{P_0}=\left(\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right)$, $\overrightarrow{N_0}=\left(\begin{array}{c}{n}_x^0\\ n_y^0\\ n_z^0\end{array}\right)$
The vector representation of the plane: $Pi:{\overrightarrow{N_1}}^T\cdot X+d_1=0$；使用$\overrightarrow{N_1}=\left(\begin{array}{c}{n}_x^1\\ n_y^1\\ n_z^1\end{array}\right)$,$d$ is a scalar.

There are three situations for the intersection point of a line to a plane:
1) On a plane
2) Parallel to a plane
3)
Intersecting

with a plane Therefore, it is necessary to determine whether the line intersects the plane. In traditional space, if a plane and a straight line are planes, the hair vectors of the straight line and the plane are perpendicular. Therefore, by calculating $\overrightarrow{N_0}$and $\overrightarrow{N_1}$angle between. If it is perpendicular to the surface the line is parallel to the plane.
$cos\left(\theta \right)->{\overrightarrow{N_0}}^T\cdot \overrightarrow{N_1}$
If it's close to 0 it means it's parallel.

The following will introduce the intersection situation.
If the plane and the line intersect, for the line, only need to calculate $t$即可。
$L:\overrightarrow{r}=\overrightarrow{P_0}+t\overrightarrow{N_0}$Put into the formula $Pi:{\overrightarrow{N_1}}^T\cdot X+d_1=0$得到如下：
$$\begin{gathered} {\overrightarrow{N_1}}^T\cdot (\overrightarrow{P_0}+t\overrightarrow{N_0})+d_1=0 \\ =>{\overrightarrow{N_1}}^T\cdot \overrightarrow{P_0}+t{\overrightarrow{N_1}}^T\cdot \overrightarrow{N_0}+d_1=0 \\ t=(-d_1-{\overrightarrow{N_1}}^T\cdot \overrightarrow{P_0})/{\overrightarrow{N_1}}^T\cdot \overrightarrow{N_0} \end{gathered}$$
Because the vector $\overrightarrow{P_0}$,$\overrightarrow{N_0}$,$\overrightarrow{N_1}$are all known, so the t = tnt=t_n that can be found$t=t_n$.
Bring in to get
$$\overrightarrow{r_n}=\overrightarrow{P_0}+t_n\overrightarrow{N_0}$$

Based on the above formula, the source code is as follows:

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;
	}