Positioning: (plücker coordination) Plücker representation and orthogonal representation of a straight line, and their mutual conversion

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foreword

The straight line in 3D space can be used by Plucker $(\mathbf{n},d)$means that it is convenient to participate in geometric operations, but its dimension is 6, and the Pluck expression is not dominant in optimization, while the orthogonal representation$(\mathbf{U},W)$has 4 degrees of freedom and is mainly involved in optimization operations. This paper mainly records the following points

• The Pluck representation of the line
• Orthogonal representation with lines
• The relationship between the two and the mutual conversion process

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1. Vector cross product and point product

The above explanation will use cross product and dot product. In order to avoid unfamiliarity, let’s first review the basic operations of vectors:
$$\begin{gathered} \mathbf{a}\cdot \mathbf{b}=b\cdot a \\ a\times b=-\mathbf{b}\times a \\ a\cdot (\mathbf{b}\times \mathbf{c})=(\mathbf{a}\times \mathbf{b})\cdot \mathbf{c} \\ a\times (b\times c)=(a\cdot \mathbf{c})\mathbf{b}-(a\cdot \mathbf{b})\mathbf{c} \end{gathered}$$

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2. Plücker coordinate system (plücker coordination)

1. Definition

In 3D space, the rigid body has 6 degrees of freedom, while the line has 4 degrees of freedom, because the line rotates around itself (roll), or translates in the direction of the line or the same straight line, so there are two less degrees of freedom.

plücker coordination was defined by Julius Plücker in the 19th century, also known as Grassmann coordinates. The Pluck coordinate system is defined as $(\mathbf{n},\mathbf{d})$，$n$ is a moment vector (moment vector) is also a normal vector,$d$ is line$The\; direction\; vector\; of\; \ell$ . Specifically: two points on the known line$p_1,p_2$, then the direction vector $\mathbf{d}$：
$$\mathbf{d}=p_1-p_2\text{\hspace{1em}( 1 )}$$
Normal vector$\mathbf{n}$：
$$\mathbf{n}=p\times d\text{\hspace{1em}(2)}$$

where $n$ perpendicular to the line$\ell$ The plane formed by the coordinate origin. $p$ is any point on the line (can be$p_1$or $p_2$) is a vector representation in world coordinates. point $The\; reason\; p$ can be chosen arbitrarily is that any point$p^{\prime }$ satisfying$p^{\prime }-p=\lambda \mathbf{d}$：
$$\begin{array}{rl}{p}^{\prime }\times \mathbf{d}& =(\mathbf{p}+(p^{\prime }-\mathbf{p}))\times d\\ & =p\times d+(p^{\prime }-\mathbf{p})\times d\\ & =\mathbf{n}+\lambda \mathbf{d}\times d\\ & =\mathbf{n}\end{array}\text{\hspace{1em}(3)}$$

2. Nature

2.1. A little $p$ onlineℓ \ellA sufficient and necessary condition on$\ell$$\mathbf{q}\times \mathbf{d}=n$ .
2.2. Coordinate origin toℓ \ellThe distance of$\ell$$\frac{||\mathbf{n}||}{||\mathbf{d}||}$.
2.3. Line $(\mathbf{n},d)$with the line$(\alpha n,\alpha \mathbf{d}),a\ne 0$ means the same line.
2.4. Line$The\; intersection\; point\; (perpendicular\; foot)\; of\; \ell$ and the vertical line passing through the origin is denoted as$p_{\perp }$,but:

$$\begin{array}{rl}{p}_{\perp }& =p-(\hat{\mathbf{d}}\cdot \mathbf{p})\hat{d}\\ & =(\hat{d}\cdot \hat{d})\mathbf{p}-(\hat{d}\cdot p)\hat{d}\\ & =\hat{d}\times (\mathbf{p}\times \hat{\mathbf{d}})\\ & =\frac{\mathbf{d}}{||\mathbf{d}||}\times (p\times \frac{d}{||d||})\\ & =\frac{d}{||d||}\times \frac{\mathbf{n}}{||d||}\\ & =\frac{d\times n}{||\mathbf{d}|{|}^2}\end{array}\text{\hspace{1em}(4)}$$

2.5. $\mathbf{n}=At\; 0$ , the straight line$\ell$ passes through the coordinate origin. Then, the Pluck representation of the line passing through the origin of coordinates is$(0,\_)$ .
2.6. Plane$\Pi$ passes through the coordinate origin, then it is located on the plane$All\; straight\; lines\; at\; infinity\; on\; \Pi$ have the same Pluck representation as$(\mathbf{n},0)$, $n$ is the surface$The\; normal\; of\; \Pi$ .
2.7. Straight line$\ell$ about any point in spaceqqThe moment vector of$q$${\mathbf{n}}_q$，$q$ inℓ \ellThe projection on$\ell$$q_{\perp }$，则：
$$\begin{array}{rl}{n}_q& =(\mathbf{p}-\mathbf{q})\times \hat{\mathbf{d}}\\ & =\mathbf{p}\times \hat{d}-q\times \hat{d}\\ & =n-q\times \hat{d}\end{array}\text{\hspace{1em}(5)}$$
又，
$$\begin{array}{rl}\hat{\mathbf{d}}\times {\mathbf{n}}_q& =\hat{d}\times ({\mathbf{q}}_{\perp }-\mathbf{q})\times \hat{d}\\ & =(\hat{d}\cdot \hat{d})({\mathbf{q}}_{\perp }-q)-(\hat{d}\cdot (q_{\perp }-\mathbf{q}))\hat{d}\\ & ={\mathbf{q}}_{\perp }-q\end{array}\text{\hspace{1em}( 6 )}$$
Therefore,
$${\mathbf{q}}_{\perp }=q+\hat{\mathbf{d}}\times {\mathbf{n}}_q\text{\hspace{1em}( 7 )}$$
the pluck representation of straight lines is suitable for geometric operations but not suitable for optimization , while the orthogonal representation of straight lines can be applied to the optimization process

3. Transform the offline Pluck coordinates of the world coordinate system to the camera coordinate system

Known transformation matrix ${\mathbf{T}}_w^c=\left[\begin{array}{cc}{\mathbf{R}}_w^c& {\mathbf{t}}_w^c\\ 0& 1\end{array}\right]$ , the straight line Pluck in the world coordinate system represents${\mathcal{L}}_w$, then Plücker in the camera coordinate system represents $L_c$Can pass formula $(8)$ 得到：
$${\mathcal{L}}_c=\left[\begin{array}{c}{\mathbf{n}}_c\\ {\mathbf{d}}_c\end{array}\right]={\mathcal{T}}_w^c{L}_w=\left[\begin{array}{cc}{\mathbf{R}}_w^c& {\left[{\mathbf{t}}_w^c\right]}_{\times }{R}_w^c\\ 0& R_w^c\end{array}\right]L_w\text{\hspace{1em}( 8 )}$$
Among them,$[t_w^c{]}_{\times }$For the translation vector ${\mathbf{t}}_w^c$The anti-symmetric matrix (Skew-symmetric matrix), $T_w^c$is the transformation matrix of the Pluck coordinates of the line from the world coordinate system to the camera coordinate system. In the same way, the transformation matrix T cw \boldsymbol{ \mathcal{T}}_c^w that transforms the straight line from the camera coordinate system to the world coordinate system${\mathcal{T}}_c^w$，如式(9)所示：
$$T_c^w={T_w^c}^T=\left[\begin{array}{cc}{{\mathbf{R}}_w^c}^T& -{\left[{{\mathbf{R}}_w^c}^T{\mathbf{t}}_w^c\right]}_{\times }{R_w^c}^T\\ 0& {R_w^c}^T\end{array}\right]\text{\hspace{1em}(9)}$$

3. Orthogonal representation of lines

Orthogonal representation requires only 4 variables $(i,)\_\in {\mathbb{R}}^3\times R^1$ , corresponding to two orthogonal matrices$(\mathbf{U},\mathbf{W})\in SO(3)\times SO\left(2\right)$ , it is more suitable for the representation of lines during optimization.

1. Convert Plücker representation to orthogonal representation

Orthogonal representation and Pluck representation can be converted to each other and can be flexibly applied in SLAM. Orthogonal representation can be obtained by QR decomposition of Plücke coordinates, but it is easy to construct the following matrix in practical applications:
$$\begin{array}{rl}\left[\mathbf{n}\mathbf{d}\right]& =\left[\begin{array}{ccc}\frac{\mathbf{n}}{||\mathbf{n}||}& \frac{\mathbf{d}}{||\mathbf{d}||}& \frac{\mathbf{n}\times \mathbf{d}}{||\mathbf{n}\times \mathbf{d}||}\end{array}\right]\left[\begin{array}{cc}||n||& 0\\ 0& ||d||\\ 0& 0\end{array}\right]\\ & =\left[\begin{array}{ccc}{\mathbf{u}}_1& u_2& u_3\end{array}\right]\left[\begin{array}{cc}||n||& 0\\ 0& ||d||\\ 0& 0\end{array}\right]\end{array}\text{\hspace{1em}(10)}$$

Therefore, it is easy to obtain an orthogonal matrix representing the rotation, that is, the rotation matrix $\mathbf{U}$：
$$\begin{array}{rl}\mathbf{U}& =R\left(i\right)={\mathbf{R}}_x(i_1){\mathbf{R}}_y\left(i_2\right)R_z(i_3)\\ & =\left[\begin{array}{ccc}{\mathbf{u}}_1& u_2& u_3\end{array}\right]\\ & =\left[\begin{array}{ccc}\frac{\mathbf{n}}{||\mathbf{d}||}& \frac{\mathbf{d}}{||d||}& \frac{\mathbf{n}\times \mathbf{d}}{||\mathbf{n}\times \mathbf{d}||}\end{array}\right]\end{array}\text{\hspace{1em}(11)}$$

where, $\theta$ is in turn around$x-y-The\; rotation\; angle\; of\; the\; z$ axis, obviously,$R\left(\theta \right)$is around$x-y-$Rotation matrix after$z\; rotation.$
Matrix$\left[\begin{array}{cc}||\mathbf{n}||& 0\\ 0& ||\mathbf{d}||\\ 0& 0\end{array}\right]$There is only one degree of freedom, which can be normalized to one $SO\left(2\right)$ equation, which has the following solution:
$$\begin{array}{rl}\mathbf{W}& =\left[\begin{array}{cc}\cos \varphi & -\sin \varphi \\ \sin & \cos \end{array}\right]\\ & =\left[\begin{array}{cc}{w}_1& -w_2\\ w_2& w_1\end{array}\right]\\ & =\frac{1}{\sqrt{||\mathbf{n}|{|}^2+||\mathbf{d}|{|}^2}}\left[\begin{array}{cc}||\mathbf{n}||& -||\mathbf{d}||\\ ||\mathbf{d}||& ||\mathbf{n}||\end{array}\right]\end{array}\text{\hspace{1em}(12)}$$

It can be seen from formula (12) that $W$ contains distance information. So the orthogonal representation$(\mathbf{U},W)$ contains 3 degrees of freedom for rotation and 1 degree of freedom for distance.

2. Orthogonal representation converted to Pluck representation

The known line d is orthogonal to represent $(U,\mathbf{W})$，或$(i,\varphi )$ , it is easy to obtain the Pluck representation:
$$\left[\mathbf{n}\mathbf{d}\right]=\left[w_1{\mathbf{u}}_1^T{w}_2{u}_2^T\right]\text{\hspace{1em}(13)}$$

Among them, $w_1=\cos \varphi ,w_2=\sin \varphi$ .

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Summarize

This article mainly records the Pluck representation and orthogonal representation of the line, as well as the mutual conversion formula between the two. Please correct me if I am wrong, thank you!

references

Jia, Yan-Bin. “Plücker Coordinates for Lines in the Space.” (2020).
He, Yijia et al. “PL-VIO: Tightly-Coupled Monocular Visual–Inertial Odometry Using Point and Line Features.” Sensors (Basel, Switzerland) 18 (2018)