Point rotation and coordinate system rotation

Point rotation transformations in the same coordinate system (1) and point transformations in different coordinate systems (2) have plagued me, they are two different concepts, but the form is very similar to the two-dimensional space as an example to do the next deduction, deepen understanding.

The point rotation transformation under the same coordinate system is better understood and is a rotational transformation in the same coordinate system. 3, it is known that the counterclockwise rotation angle is θ, we introduce the intermediate variable vector length r and the horizontal angle α, obviously, the derivation formula is as follows:

The homogeneous coordinate system is expressed as:

point transformations in different coordinate systems, which are commonly used in perspective transformations, are useful for mapping a point from one coordinate system to another, which is helpful in mapping the world coordinate system to a polar coordinate system. As shown in 4, the known coordinate system o ' x ' y ' rotation angle of θ,o ' x ' y ' relative to the Oxy coordinate system is the coordinate Origin o ' with respect to oxy coordinates (X0,Y0), we introduce the intermediate variable vector length r and the horizontal angle α. The idea of the transformation is to first rotate θ on the O ' X ' Y ' coordinate system and then pan (X0,Y0). The derivation process is as follows:

The homogeneous coordinate system is expressed as:

Note that the function of the homogeneous coordinates is to combine the rotation and zoom and move together, in the traditional Euclidean space can not be done, it is necessary in the projection space in the homogeneous coordinate system to complete.

The same can be extended to three-dimensional space. OXYZ coordinate system can be regarded as the camera coordinate system, O ' X ' Y ' Z ' can be regarded as the world coordinate system,

Resources:

[1]. Coordinate transformation of the matrix (rotation) (which describes the matrix's rotational scaling, as well as the derivation process, highly recommended ★★★★★)

Point rotation and coordinate system rotation