Hessian Normal Form Plane Equations

Plane Equations

It is especially convenient to specify planes in so-called Hessian normal form. This is obtained from the general equation of a plane

以所谓的 Hessian 法线形式指定平面尤其方便。这是从平面的一般方程中得到的

\[ax + by + cz + d = 0 \tag{1}\]

by defining the components of the unit normal vector $\mathbf{n} = (n_x,\ n_y,\ n_z)$:

通过定义单位法向量的分量 $\mathbf{n} = (n_x,\ n_y,\ n_z)$：

\[n_x = \frac{a}{\sqrt{a^2+b^2+c^2}} \tag{2}\] \[n_y = \frac{b}{\sqrt{a^2+b^2+c^2}} \tag{3}\] \[n_z = \frac{c}{\sqrt{a^2+b^2+c^2}} \tag{4}\]

and the constant

和常量

\[q = \frac{d}{\sqrt{a^2+b^2+c^2}} \tag{5}\]

as well as point $\mathbf{p}=(p_x,\ p_y,\ p_z)$ in the plane. Then the Hessian normal form of the plane is

以及平面内的点 $\mathbf{p}=(p_x,\ p_y,\ p_z)$。然后平面的海森法线形式是

\[\mathbf{n} \cdot \mathbf{p} = -q \tag{6}\]

note that $\mathbf{n} \cdot \mathbf{p}$ is dot product of vectors, and $q$ is the distance of the plane from the origin. Here, the sign of $q$ determines the side of the plane on which the origin is located. If $q>0$, it is in the half-space determined by the direction of $\mathbf{n}$, and if $q<0$, it is in the other half-space.

注意 $\mathbf{n} \cdot \mathbf{p}$ 是两向量的点积，并且 $q$ 是从原点到平面的距离。$q$ 的符号决定了原点是位于平面的哪一边。如果 $q>0$，原点是在 $\mathbf{n}$ 的方向确定的一半空间内，如果 $q<0$，原点是在另一半空间内。

The point-plane distance from a point $\mathbf{p}_0$ to a plane (6) is given by the simple equation

从一点 $\mathbf{p}_0$ 到平面 (6) 的点-面距离由简单的方程给出

\[D = \mathbf{n} \cdot \mathbf{p}_0 + q \tag{7}\]

If the point $\mathbf{p}_0$ is in the half-space determined by the direction of $\mathbf{n}$, then $D>0$; if it is in the other half-space, then $D<0$.

如果点 $\mathbf{p}_0$ 在 $\mathbf{n}$ 的方向确定的一半空间里，那么 $D>0$；如果它在另一半空间里，那么 $D<0$。

涉及的向量与矩阵知识点

References

• Plane — Wolfram Mathworld

• Hessian Normal Form — Wolfram Mathworld

• Point-Plane Distance — Wolfram Mathworld