How to understand plucker coordinates (geometric understanding)

目标：How to understand the plucker coordinates
Recent review articles have linear expression.Because recently in writing project,Encountered this kind of problem,But this kind of problem,Few Chinese information,Therefore his record yourself understand.

定义： The traditional in any of several European coordinates,直线的向量表达：$L:\overrightarrow{r}=\overrightarrow{\mathbf{P}}+t\overrightarrow{\mathbf{l}}$;其中$\overrightarrow{\mathbf{P}}=\left(\begin{array}{c}{p}_0\\ p_1\\ p_2\end{array}\right)$, $\overrightarrow{\mathbf{l}}=\left(\begin{array}{c}{l}_0\\ l_1\\ l_2\end{array}\right)$, 其中$\overrightarrow{\mathbf{P}}$Vertex said line,$\overrightarrow{\mathbf{l}}$Is the direction of the straight line.

定义： In the plucker coordinate system,Linear expression generally represented as$(\overrightarrow{\mathbf{l}},\overrightarrow{\mathbf{m}})$,其中$\overrightarrow{\mathbf{m}}=\overrightarrow{\mathbf{p}}\times \overrightarrow{\mathbf{l}}$,称为：$momentvector$, $\overrightarrow{\mathbf{l}}$Is the direction of the straight line.

But a lot of people not particularly understand$(\overrightarrow{\mathbf{l}},\overrightarrow{\mathbf{m}})$On the geometric meaning expression in a straight line.And it and a few in any coordinate system of the straight line have what relation？

答：
You first need to understand a very important vector$\overrightarrow{\mathbf{l}}$,It is a direction of the straight line.It and in any of several European straight direction,Said straight direction.如下图,Is the direction of the straight line,Generally do not have the length information,可以normlize,But if it doesn'tnormlize,它提供一个scale的量,可以表示为$c=|\overrightarrow{\mathbf{l}}|$.而这个$c$There are provided$\overrightarrow{\mathbf{m}}$ （因为$\overrightarrow{\mathbf{m}}$也来源于$\overrightarrow{\mathbf{l}}$）. Through the calculation of later found$\overrightarrow{\mathbf{l}}$不提供scale变量,其实scaleVertex variable from line$\overrightarrow{\mathbf{P}}$. $\overrightarrow{\mathbf{P}}$和$\overrightarrow{\mathbf{m}}$ Jointly decided the another important variable of linear,The apex of the straight line.Below is expressed as$\overrightarrow{\mathbf{P}}\perp$,It is perpendicular to the vertex of the straight line.

Understand another very important vector again$\overrightarrow{\mathbf{m}}$,It is besides said direction,It also has a length of information,And it and vertex in line$\overrightarrow{\mathbf{p}}$无关.这是一个非常重要的性质.
证明：$\overrightarrow{\mathbf{m}}$和$\overrightarrow{\mathbf{p}}$(Any vertex in line)无关,假设$\overrightarrow{\mathbf{m}}=\overrightarrow{\mathbf{p}}\times \overrightarrow{\mathbf{l}}$ ;The apex of the straight line on the other $\overrightarrow{{\mathbf{p}}^{\prime }}$For straight line of arbitrary vertex,$\overrightarrow{{\mathbf{p}}^{\prime }}-\overrightarrow{\mathbf{p}}=\lambda \overrightarrow{\mathbf{l}}$.Prove vertex and line has nothing to do,因此得到如下：
$$\begin{gathered} \overrightarrow{{\mathbf{p}}^{\prime }}\times \overrightarrow{\mathbf{l}} \\ =(\lambda \overrightarrow{\mathbf{l}}+\overrightarrow{\mathbf{p}})\times \overrightarrow{\mathbf{l}} \\ =\lambda \overrightarrow{\mathbf{l}}\times \overrightarrow{\mathbf{l}}+\overrightarrow{\mathbf{p}}\times \overrightarrow{\mathbf{l}} \\ =\overrightarrow{\mathbf{p}}\times \overrightarrow{\mathbf{l}}=\overrightarrow{\mathbf{m}} \end{gathered}$$

So you can see on the drawing：$\overrightarrow{\mathbf{m}}$是垂直于$\overrightarrow{\mathbf{p}}$和$\overrightarrow{\mathbf{l}}$所在的平面.$\overrightarrow{\mathbf{m}}$的长度表示为$|\overrightarrow{\mathbf{m}}|$.Vertical point is expressed as the origin to a straight line$\overrightarrow{\mathbf{P}}\perp$.
如果向量$\overrightarrow{\mathbf{l}}$Is a unit vector.向量$\overrightarrow{\mathbf{P}}\perp$的长度就和$|\overrightarrow{\mathbf{m}}|$长度一样,It is also a vertex positioning line,Equivalent to a linear one vertex(Therefore linear direction and vertex,The only decision in a straight line).为了证明这个结论,假设$\overrightarrow{{\mathbf{l}}^{\prime }}$A unit vector for the line,因为$\overrightarrow{\mathbf{m}}$来源于$\overrightarrow{\mathbf{p}}\times \overrightarrow{\mathbf{l}}$,So it will change,可以表示为$\overrightarrow{{\mathbf{m}}^{\prime }}$.因此：c$\overrightarrow{{\mathbf{l}}^{\prime }}=\overrightarrow{\mathbf{l}}$;c$\overrightarrow{{\mathbf{m}}^{\prime }}=\overrightarrow{\mathbf{m}}$

计算$\overrightarrow{\mathbf{P}}\perp =\overrightarrow{{\mathbf{l}}^{\prime }}\times \overrightarrow{{\mathbf{m}}^{\prime }}$

Expression vector as follows：$\overrightarrow{\mathbf{P}}\perp \overrightarrow{\mathbf{P}}=t\overrightarrow{{\mathbf{l}}^{\prime }}$, $\overrightarrow{\mathbf{O}}\overrightarrow{\mathbf{P}}=\overrightarrow{\mathbf{P}}$(因为O是原点).

因为知道$cos(\theta )=|\overrightarrow{\mathbf{P}}\perp \overrightarrow{\mathbf{P}}|/|\overrightarrow{\mathbf{O}}\overrightarrow{\mathbf{P}}|=(\overrightarrow{{\mathbf{l}}^{\prime }}\cdot \overrightarrow{\mathbf{P}})/(|\overrightarrow{{\mathbf{l}}^{\prime }}|\cdot |\overrightarrow{\mathbf{P}}|)$,This is the basic of triangle function formula.

$$\begin{gathered} |\overrightarrow{\mathbf{P}}\perp \overrightarrow{\mathbf{P}}| \\ =|\overrightarrow{\mathbf{O}}\overrightarrow{\mathbf{P}}|cos(\theta ) \\ =(\overrightarrow{{\mathbf{l}}^{\prime }}\cdot \overrightarrow{\mathbf{P}})/|\overrightarrow{{\mathbf{l}}^{\prime }}| \\ =\overrightarrow{{\mathbf{l}}^{\prime }}\cdot \overrightarrow{\mathbf{P}}(因为\overrightarrow{{\mathbf{l}}^{\prime }}是单位向量) \end{gathered}$$

Calculate the formula above,Vertical point can be as follows：

$$\begin{gathered} \overrightarrow{\mathbf{P}}\perp =\overrightarrow{\mathbf{P}}-(\overrightarrow{{\mathbf{l}}^{\prime }}\cdot \overrightarrow{\mathbf{P}})\overrightarrow{{\mathbf{l}}^{\prime }} \\ =(\overrightarrow{{\mathbf{l}}^{\prime }}\cdot \overrightarrow{{\mathbf{l}}^{\prime }})\overrightarrow{\mathbf{P}}-(\overrightarrow{{\mathbf{l}}^{\prime }}\cdot \overrightarrow{\mathbf{P}})\overrightarrow{{\mathbf{l}}^{\prime }} \\ =\overrightarrow{{\mathbf{l}}^{\prime }}\times (\overrightarrow{\mathbf{P}}\times \overrightarrow{{\mathbf{l}}^{\prime }}) \\ =\overrightarrow{{\mathbf{l}}^{\prime }}\times \overrightarrow{{\mathbf{m}}^{\prime }} \end{gathered}$$

Can be seen from the formula,The apex of the straight line$\overrightarrow{\mathbf{P}}\perp$The two formula$\overrightarrow{{\mathbf{l}}^{\prime }}\times \overrightarrow{{\mathbf{m}}^{\prime }}$得到.It also determines the uniqueness of a straight line.$\overrightarrow{\mathbf{P}}\perp$变化,Although the direction,But it's position will change,Also change the location of the said a straight line.Vertex so line,由$\overrightarrow{{\mathbf{l}}^{\prime }}\times \overrightarrow{{\mathbf{m}}^{\prime }}$决定.

如果$\overrightarrow{\mathbf{l}}$不是单位向量,It has its own length,因为$\overrightarrow{\mathbf{l}}$Units can be said to$\overrightarrow{{\mathbf{l}}^{\prime }}=\overrightarrow{\mathbf{l}}/|\overrightarrow{\mathbf{l}}|=\overrightarrow{\mathbf{l}}/c$,因此公式中:
$$\begin{gathered} \overrightarrow{\mathbf{P}}\perp =\overrightarrow{{\mathbf{l}}^{\prime }}\times \overrightarrow{{\mathbf{m}}^{\prime }}=(\overrightarrow{\mathbf{l}}/|\overrightarrow{\mathbf{l}}|)\times \overrightarrow{{\mathbf{m}}^{\prime }} \\ =(\overrightarrow{\mathbf{l}}\times \overrightarrow{{\mathbf{m}}^{\prime }})/|\overrightarrow{\mathbf{l}}|=(\overrightarrow{\mathbf{l}}\times \overrightarrow{{\mathbf{m}}^{\prime }})/c \\ =(\overrightarrow{\mathbf{l}}\times (\overrightarrow{\mathbf{m}}/c))/c=(\overrightarrow{\mathbf{l}}\times \overrightarrow{\mathbf{m}})/c^2 \end{gathered}$$

从公式可以看出来：
原点到直线的距离,可以表示为$|\overrightarrow{\mathbf{P}}\perp |=|(\overrightarrow{\mathbf{l}}\times \overrightarrow{\mathbf{m}})|/c^2=|(\overrightarrow{\mathbf{l}}\times \overrightarrow{\mathbf{m}})|/|\overrightarrow{\mathbf{l}}{|}^2$,可以得到,无论$\overrightarrow{\mathbf{l}}$是否normlize,只要$\overrightarrow{\mathbf{m}}$和 normlized之后的$\overrightarrow{\mathbf{l}}$保持一致,都可以.

同时可以看出来
$$(\overrightarrow{\mathbf{l}},\overrightarrow{\mathbf{m}})=(c\overrightarrow{\mathbf{l}},c\overrightarrow{\mathbf{m}})$$
Equal sign says that they are the same linear expression.
证明：直接代入$\overrightarrow{\mathbf{P}}\perp$公式就行了.

参考论文：Plücker Coordinates for Lines in the Space ∗