The derivation of transformation matrix between 1 and three-dimensional coordinate system of graphic science

Summary: The transformation of the three-dimensional coordinate system is essentially the change of the origin and the orthogonal base vector, which is represented as translation and rotation in space.

The transformation of the coordinate system as shown in the figure can be represented by a transformation matrix.

Although the principle is relatively simple, but the freshman linear algebra has been a little forgotten. =////=

Next, as a refresher, let me deduce how this transformation matrix is derived, using the knowledge of some relatively basic linear algebra.

First of all, we need to understand why this matrix is needed, and under what circumstances we need this matrix. Obviously, if we need to convert the coordinates of the original coordinate system to the coordinates in the new coordinate system, we can use this matrix in the case of the base vector of the new coordinate system known.

Known: The coordinates of the point P in the original coordinate system are (x, Y, z), the unit vector for the y direction is, the unit vector for the z direction is the unit vector for the axis,

The origin of the new coordinate system is in the coordinates of the original coordinate system (X0,Y0,Z0),

The unit vector for the x-axis direction is

The unit vector for the y-axis is

The unit vector for the z-axis direction is

The coordinates of the point P in the new coordinate system (x ', y ', Z ').

Understanding: The so-called point coordinates are actually the linear combination of vectors, and the transformation of the base vector between the coordinate system is also a linear combination of vectors, it is obvious to think of matrices.

In order to get this matrix, we need to construct a linear equation set. As follows:

According to this equation group we can get it more easily:

Next is the last step, I asked for the inverse matrix of this square, here only need to first left multiply a matrix to make the matrix into this form.

Then, a, B is the inverse of the matrix can be, and a matrix is an orthogonal matrix, the inverse matrix of the orthogonal matrix equals the transpose matrix, here only need to simply transpose, and b matrix is [1], the inverse matrix is [1].

Finally get,

The square on the right is the transformation matrix we need.

Summary: The transformation of the coordinate system, in fact, is to show the measurement standard through another set of orthogonal base vectors, and the transformation of the orthogonal base vector is the linear combination of the original orthogonal base vectors.

by-Shan Book 2017-4-8