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

目的：最近写C++代码,遇到一些基础的算法.Such as calculating the distance from the vertex to the plane.

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

计算$P_1$到平面$Plane$的距离.
$Plane$expression can be converted to $n_0(x-x_0)+n_1(y-y_0)+n_2(z-z_0)=0$,计算得到$n_{0}x+n_{1}y+n_{2}z+d=0$其中$d=-n_0{x}_0-n_1{y}_0-n_2{z}_0=-N^T\cdot P_0$

推导公式如下：

Calculate the red vertices$P_1$to the blue plane$Plane$的距离.The yellow vertex is a point on the plane$P_0$.Vertex-to-plane distance definition：The distance from the point where it is perpendicular to the plane（Represented by grey line segments in the figure）.

The algorithm based on the vertical triangle obtains the length of the gray line segment as $dist$
$$dist=\Vert P_0-P_1\Vert \cdot cos(\theta )(1)$$
其中$\theta$是两个向量$P_0{P}_1$and the plane normal vector$N$的夹角,
$$cos(\theta )=\frac{|N^T\cdot (P_0-P_1)|}{\Vert P_0-P_1\Vert \cdot \Vert N\Vert }(2)$$
将公式(2)带入公式(1)可以得到如下:
$$dist=\frac{|N^T\cdot (P_0-P_1)|}{\Vert N\Vert }(3)$$
将向量$P_0=\left(\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right)$,$P_1=\left(\begin{array}{c}{x}_1\\ y_1\\ z_1\end{array}\right)$,$N=\left(\begin{array}{c}{n}_0\\ n_1\\ n_2\end{array}\right)$带入公式(3)the resulting 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 in algebraic form as :
$$dist=\frac{|n_0{x}_1+n_1{y}_1+n_2{z}_1+d|}{\sqrt{n_0^2+n_1^2+n_2^2}}(5)$$

The source code is as follows,It is written in vector based schema.It needs to be configured by youEigen库：

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