Analysis of the properties of rotation matrix (Rotate matrix)

　Having studied matrix theory or linear algebra definitely knows that the orthogonal matrix (orthogonal matrix) is a very good matrix, why do you say so? Here are a few reasons:

1. Each column of an orthogonal matrix is a unit matrix, and 22 intersects. The simplest orthogonal matrix is the unit array.
2. The inverse of the orthogonal matrix (inverse) equals the transpose of the orthogonal matrix (transpose). At the same time, it can be inferred that the value of the determinant of the orthogonal matrix is positive or negative 1 .
3. The orthogonal matrix satisfies many matrix properties, for example, it can be similar to the diagonal matrix and so on.

The above can be seen that the orthogonal matrix is a very special matrix, and the rotation matrix in this topic is an orthogonal matrix! It perfectly interprets all the characteristics of the orthogonal matrix.

First of all, what is a rotation matrix? As shown in 1, we assume that the coordinate system of the first spaceXA，YA，Zis the Cartesian coordinate system, so we get the matrix of space A.VA ={XA ，YA ，ZA }t, in fact, can also be seen as a unit arrayE。 After the rotation, the three coordinate system of space A becomes the three coordinate system of the red in Figure 1XB ，YB ，ZB , get the matrix of space BVB ={XB ，YB ，ZB }t. We connect two spaces to get aVB =R•VA Over hereRis what we call the rotation matrix. BecauseXA ={1,0,0}t,YA ={0,1,0}t,ZA ={0,0,1}t, as can be seen in Figure 2, the rotation matrixRis theXB ，YB ，ZB Consisting of three vectors. In this case, you should find the rotation matrixRSatisfies the first condition, because the unit vector does not change regardless of the length of the rotation and the orthogonal properties between the vectors do not change. SoThe rotation matrix is the orthogonal array！ But this does not explain the problem, I further use the mathematical formula to prove.

Fig. 1 Fig. 2

Before we talk about the general situation, let's start with two points of mathematical knowledge. (1) The geometric meaning of the dot product: 3, we can get the Shan β equivalent to the β modulus on β by the formula of the point multiplication, so when | When β|=1,Shan β refers to the modulo of α projecting on β . This is important in the following sections. (2) The geometrical meaning of inverse of rotation matrix: This is more abstract, but understanding. The rotation matrix is equivalent to the rotation of a vector (space) into a new vector (space), then the inverse can be understood as a new vector (space) back to the original vector (space).

Figure 3

　　 the next thing is the point., we analyze it in conjunction with Figure 4. As explained above, the rotation matrixRis theXB ，YB ，ZB Consisting of three vectors. Let's take a look.XB ，YB ，ZB What the hell is that? Since all vectors in the graph are unit vectors,XB AndXA The result of the point multiplication can be seen asXB InXA On the projection of the model, namelyXB In Space acomponent of the x-axis！！ The middle position in the diagram lists theXB The three components in a vector areXB InXA The projection on the module,XB InYA The model and the projection on theXB InZA The projection on the module. This is well understood from a geometrical point of view. And so on, you can derive the rotation matrixRform of expression. We can be surprised by the figure 4 that the first line of Matrix R isXA InXB ，YB ，ZB On the projection of the model, namelyX at。What's the use of this discovery? Figure 5 explains. According to the above formula can be introduced a to b rotation matrix equals b to a rotation of the matrix transpose. According to what we said in the last paragraph, the inverse of a to B rotation matrix is equal toBto theAthe rotation matrix, so it's easy to roll out R-1 equals RT! This satisfies the second condition of the orthogonal matrix, and proves again that the rotation matrix is an orthogonal array . In peacetime work, I also tested all the rotation matrix determinant of the value is 1, so the rotation matrix to meet the orthogonal array of all the properties, can be said to be a perfect matrix.

　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　　

Fig. 4 Fig. 5  

give some Lezilai tomorrow to explain the nature of the rotation matrix, come first.

　　

Analysis of the properties of rotation matrix (Rotate matrix)