Rotation matrix derivation process

Why is the rotation matrix a set of many sine and cosine function values , What are the properties of rotation matrix ？ What is the meaning of the rotation matrix ？ What is the difference between the expressions of two-dimensional rotation and three-dimensional rotation ？

In this paper, firstly, the rotation matrix algebra and matrix expression in two-dimensional case are derived , Then the results are extended to three-dimensional .

One 、 Rotation matrix in two-dimensional space

What if $P$ In the coordinate system $\{B\}$ The coordinates of the for $P=\{x,y\}$, There is another coordinate system $\{A\}$, Can you give the same point $P$ stay $\{A\}$ Coordinate values in a coordinate system $P^{\prime }=\{x^{\prime },y^{\prime }\}$？ The answer is yes , The premise is that these two coordinate systems are known $\{A\}$$\{B\}$ The relationship between , When the relationship between two coordinate systems is pure rotation , Rotation matrix is an expression used to describe the relationship between two coordinate systems .（ Simply speaking , Is known $\{B\}$ Point coordinates under , seek $\{A\}$ Point coordinates under ). Here's the picture ：

$\{A\}$ The coordinate system rotates counterclockwise $\theta$ Degree and $\{B\}$ The winning system completely overlaps , For ease of description , Let's assume the origin to $P$ The distance to $r$, Consider the coordinate system $\{B\}$, According to the definition of rectangular coordinate system ：
$$\begin{gathered} x=r\cos \alpha \\ y=r\sin \alpha \text{\hspace{1em}(1)} \end{gathered}$$

Consider the coordinate system $\{A\}$, Yes ：
$$\begin{gathered} x^{\prime }=r\cos (\alpha +\theta ) \\ {y}^{\prime }=r\sin (\alpha +\theta )\text{\hspace{1em}(2)} \end{gathered}$$
The expanded expression is ：
$$\begin{gathered} x^{\prime }=r(\cos \alpha \cos \theta -\sin \alpha \sin \theta ) \\ {y}^{\prime }=r(\sin \theta \cos \alpha +\cos \theta \sin \alpha )\text{\hspace{1em}(3)} \end{gathered}$$
take （1） Into the （3） have to ：
$$\begin{gathered} x^{\prime }=x\cos \alpha -y\sin \alpha \\ {y}^{\prime }=y\cos \alpha +x\sin \alpha \text{\hspace{1em}(4)} \end{gathered}$$
（4） That's what we're looking for , Realize the algebraic expression of the same point in different coordinate systems , This expression is completed by $\{A\}$ Indicates that the dot arrives $\{B\}$ Transformation of points in coordinate system . Write it in matrix form ：
$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]\text{\hspace{1em}(5)}$$
Note that the matrix formed by the sine and cosine values in the above formula is ${}_A^{B}R$, By $\{A\}$ Switch the coordinate system to $\{B\}$ The rotation matrix of , about $\{B\}$ All points can be obtained by left multiplying the rotation matrix . remember ${}^{B}P$ For the coordinate system $\{B\}$ Next point ,${}^{A}P$ For the coordinate system $\{A\}$ Next point , therefore （5） Can be written as ：
$${}^{A}P{=}_B^A{R}^{B}P\text{\hspace{1em}(6)}$$
So the expression of two-dimensional rotation matrix is ：
$${}_B^{A}R=\left[\begin{array}{cc}\cos \alpha & -\sin \alpha \\ \sin \alpha & \cos \alpha \end{array}\right]\text{\hspace{1em}(7)}$$

Two 、 Rotation matrix in three-dimensional space

For two-dimensional space , Rotate around $X\times Y$ Direction of rotation , Observe the rotation behavior from the opposite direction of the arrow , Clockwise angle is positive , Counter clockwise is negative , Pure rotation in three-dimensional space , There will be three situations , Along the coordinate system $X$$Y$$Z$ Observe the rotation in the opposite direction , The same is “ Positive and negative ”. The following is the plan view of rotating a certain angle in three cases ：



In the first , Around the $Z$ Take rotation as an example , Explain its rotation matrix expression . Suppose the point before rotation is ${}^{OXY}P=\{x,y,z\}$, The point after rotation is marked as ${}^{O{X}^{\prime }{Y}^{\prime }}P=\{x^{\prime },y^{\prime },z^{\prime }\}$, The original coordinate system rotates to the new coordinate system $\theta$ Completely coincide with the new coordinates , It is similar to the derivation of rotation matrix in two-dimensional space ：
$$\begin{array}{rl}{x}^{\prime }& =x\cos \alpha -y\sin \alpha \\ y^{\prime }& =y\cos \alpha +x\sin \alpha \\ z^{\prime }& =z\end{array}\text{\hspace{1em}(8)}$$
In matrix form ：
$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right]=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right]\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\\ z^{\prime }\end{array}\right]\text{\hspace{1em}(9)}$$
Remember that the rotation matrix is $rotz$：
$$rotz(\theta )=\left[\begin{array}{ccc}\cos \alpha & -\sin \alpha & 0\\ \sin \alpha & \cos \alpha & 0\\ 0& 0& 1\end{array}\right]\text{\hspace{1em}(10)}$$
In the same way, we can get $roty$ and $rotx$：
$$roty(\theta )=\left[\begin{array}{ccc}\cos \theta & 0& \sin \theta \\ 0& 1& 0\\ -\sin \theta & 0& \cos \theta \end{array}\right]\text{\hspace{1em}(11)}$$

$$rotx(\theta )=\left[\begin{array}{ccc}1& 0& 0\\ 0& \cos \theta & \sin \theta \\ 0& -\sin \theta & \cos \theta \end{array}\right]\text{\hspace{1em}(12)}$$

3、 ... and 、 Point rotation corresponds to rotation matrix

What I mentioned earlier is the change of point coordinates caused by coordinate system switching , If a point in space really rotates around an axis , So how should this point change ？ Here's the picture ：

$P(x,y)$ The dot rotates counterclockwise to $P^{\prime }(x^{\prime },y^{\prime })$,, There is an equation before the rotation ：
$$\begin{gathered} x=r\cos \alpha \\ y=r\sin \alpha \text{\hspace{1em}(13)} \end{gathered}$$
After rotation , radius $r$ No change ：
$$\begin{gathered} x^{\prime }=\cos (\alpha +\theta ) \\ {y}^{\prime }=\sin (\alpha +\theta )\text{\hspace{1em}(14)} \end{gathered}$$
After expansion, there are ：
$$\begin{gathered} x^{\prime }=r\cos \alpha \cos \theta -rsin\alpha \sin \theta \\ {y}^{\prime }=r\sin \alpha \cos \theta +r\cos \alpha \sin \theta \text{\hspace{1em}(15)} \end{gathered}$$
simultaneous equation （13）（15） Yes ：
$$\begin{gathered} x^{\prime }=x\cos \theta -y\sin \theta \\ {y}^{\prime }=y\cos \theta +x\sin \theta \text{\hspace{1em}(16)} \end{gathered}$$
In the form of a matrix ：
$$\left[\begin{array}{c}{x}^{\prime }\\ y^{\prime }\end{array}\right]=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]\left[\begin{array}{c}x\\ y\end{array}\right]\text{\hspace{1em}(17)}$$
Make :
$$R=\left[\begin{array}{cc}\cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end{array}\right]$$
（17） It can be written. ：
$$P^{\prime }=RP\text{\hspace{1em}(18)}$$
Because only one coordinate system is involved , So this matrix coordinates without any letters , This matrix can help us calculate the coordinates of the points after rotation , It is also known as the rotation matrix , It is exactly the same as the rotation expression of the coordinate system , The rotation matrix of the actual rotation of three-dimensional points , There is no further derivation here .

Four 、 The meaning of rotation matrix

There are three meanings of rotation matrix ：

• Describe the relationship between coordinate systems , The three principal axes of the coordinate system project to the three principal axes of another coordinate system ;
• Transformation left multiplication matrix of the same point in different coordinate systems ;
• The transformation left multiplication matrix in the same coordinate system after point rotation ;

If there is only rotation relationship between two coordinate systems , Then this relationship can be expressed by the rotation matrix , This relationship is directional , That is to say $\{A\}$ Coordinate system to $\{B\}$ It's not equal to $\{A\}$ To $\{B\}$, But there are the following relationships ：
$${}_B^{A}R{=}_A^B{R}^{-1}\text{\hspace{1em}(19)}$$
And because the rotation matrix is a unit orthogonal matrix , So there is ：
$${}_B^{A}R{=}_A^B{R}^T\text{\hspace{1em}(20)}$$
In practical use , This directionality should be made clear , If you know $\{B\}$ A point in a coordinate system ${}^{B}P$ Ask for in $\{B\}$ Next point , Then rotate the matrix $\theta$ A variable's value , Should be $\{A\}$ How many angles does the coordinate system rotate in which direction , If it's clockwise , Then this value should be negative , On the contrary, it is a positive number .

Tips： How to determine the positive and negative of the rotation matrix ？ For the rotation matrix of the actual rotation type of the point , It should be how many degrees of counterclockwise rotation to reach the new point before rotation ; For the rotation matrix of points in different coordinate systems , It should be how the target coordinate system rotates to the coordinate system of the current point .