Proof and source code of distance calculation from space vertex to plane

Purpose: I recently wrote C++ code and encountered some basic algorithms. Such as calculating the distance from a vertex to a plane.

The previous article has introduced the plane and its mathematical expression:
geometric expression $Plane$ : plane normal vector$N=\left(\begin{array}{c}{n}_0\\ n_1\\ n_2\end{array}\right)$, and a vertex $P_0=\left(\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right)$
Vertex expression: $P_1=\left(\begin{array}{c}{x}_1\\ y_1\\ z_1\end{array}\right)$

Calculate $P_1$To plane $Planedistance$ .$\_$$\_$$\_$
$The\; expression\; of\; Plane$ can be transformed into$n_0(x-x_0)+n_1(y-y_0)+n_2(z-z_0)=0$ , calculate$n_{0}x+n_{1}y+n_{2}z+d=0$其中$d=-n_0{x}_0-n_1{y}_0-n_2{z}_0=-N^T\cdot P_0$

The derivation formula is as follows:

Calculate the red vertex $P_1$To the blue plane $Planedistance$ .$\_$$\_$$\_$The yellow vertex is a point P 0 P_0on the plane$P_0$. Vertex-to-plane distance definition: the distance of its point perpendicular to the plane (indicated by the gray line segment in the figure).

Based on the vertical triangle algorithm, the length of the gray line segment is $dist$
$$dist=\Vert P_0-P_1\Vert \cdot cos\left(\theta \right)\left(1\right)$$
increase$\theta$ are two vectors$P_0{P}_1$and plane normal vector $For\; example$ ,
$$cos\left(i\right)=\frac{|N^T\cdot (P_0-P_1)|}{\Vert P_0-P_1\Vert \cdot \Vert N\Vert }\left(2\right)$$
Substituting formula (2) into formula (1) can be obtained as follows:
$$dist=\frac{|N^T\cdot (P_0-P_1)|}{\Vert N\Vert }\left(3\right)$$
Put the vector$P_0=\left(\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right)$,$P_1=\left(\begin{array}{c}{x}_1\\ y_1\\ z_1\end{array}\right)$,$N=\left(\begin{array}{c}{n}_0\\ n_1\\ n_2\end{array}\right)$The equation
$$dist=\frac{|N^T\cdot P_0-N^T\cdot P_1|}{\Vert N\Vert }=\frac{|-d-N^T\cdot P_1|}{\Vert N\Vert }=\frac{|N^T\cdot P_1+d|}{\Vert N\Vert }(4)$$

The above vector form can be rewritten into an algebraic form:
$$dist=\frac{|n_0{x}_1+n_1{y}_1+n_2{z}_1+d|}{\sqrt{n_0^2+n_1^2+n_2^2}}(5)$$

The source code is as follows, it is written based on vector mode. It requires you to configure the Eigen library:

float Pnt3d2PlaneDist(Eigen::Vector4f& fn, Eigen::Vector3f& pnt)
	{
    
    
		Eigen::Vector3f n = Eigen::Vector3f(fn[0], fn[1], fn[2]);
		float sd = n.norm();
		if (sd < 1e-8)
			return std::numeric_limits<float>::max();
		float sb = std::abs(n.dot(pnt) + fn[3]);
		float d = sb / sd;
		return d;
	}

See the website for details: https://mathinsight.org/distance_point_plane