Detailed description of Affine Transformation

As an important transformation in two-dimensional plane, affine transform is widely used in the field of image graphics. Many people do not have a sensory understanding of "affine". I think it is necessary to first talk about "affine ".

The so-called "affine transformation" is a simple transformation. Its changes include rotation, translation, and scaling. After the original linear affinine transformation, it is still a straight line, the original parallel line is still a parallel line after the affine transformation, which is an affine.

The matrix of the affine transformation is the transformation matrix of the Second coordinate form

The transformation contained in this matrix involves rotation and moving, which is actually a mixture of two matrices, manyArticleAll of them are described in detail. In the mathematical formula of affine transformation, how can we achieve the translation of coordinate points? This is the key to understanding the affine transformation.

Here is a very important figure, which is shown in Baidu encyclopedia. This figure can be used to deduce the formula of the affinine transformation. The derivation is as follows:

The coordinate of a point P in the original coordinate system is (xsp, YSP ). Then we need to complete the rotation operation. The rotation operation is based on the origin. Here we use a technique to get the coordinates of the rotated points,The rotation of a point in the coordinate system can be equivalent to derotating the coordinate axis.So there is a coordinate system in which the dotted line centered on (xs0, ys0) is perpendicular to the screen horizontally. Determining the coordinates of P in this coordinate system is equivalent to determining the coordinates of P after rotation in the blue coordinate system. Based on this conclusion, we can determine the coordinates of P in the new coordinate system through simple Three-dimensional Geometric knowledge. The X and Y coordinates of P in the new coordinate system are

The classic affinine transform model is coming soon. Sort out the above two statements:

This is the principle of rotating part in the affine transform model. Another step is translation.

After the rotation and transformation, we have determined the position of the point in the new coordinate system, and then added its offset on the X and Y axes to the position.

The matrix of the ry transform is born. Of course, the processing of this transformation is more clever. It still uses the strategy of moving the coordinate system by moving the coordinate system to the opposite direction. As a result, we can see the graph of this classical affinine transformation model. We can see that the position of the point is not moving during the process of the point-to-point affine transformation. We indirectly achieve the point-to-point moving effect through constant coordinate system adjustment, this fully demonstrates one thing: Exercise is relative. Matrix theory is a profound embodiment of this philosophy of motion. If you are interested, read this article http://blog.csdn.net/xiaojidan2011/article/details/8213873.

Other references:

Http://www.cnblogs.com/shijibao001/articles/1225962.html