Affine transformation and homogeneous coordinates

Affine transformation (affine transformation) homogeneous coordinate system (homogeneous coordinate) definition:

The so-called linear transformation refers to the mapping of two linear spaces, a transformation is a linear transformation, must meet two conditions, that is, we often say the linear condition:

Additivity

Homogeneity

Understand:

In "3D Math Basics: Graphics and game development" 9.4.2 the 4x4 translation matrix, because the 3x3 matrix translation is not available

Multiplication means, in other words, we vector to represent a point in space:

R = {Rx,ry,rz}

and the translation vector is

t = {Tx,ty,tz}

Then the generalization approach is:

R + t = {rx+tx, ry+ty, Rz+tz}

The so-called multiplication means:

r + t = R. X (x = unknown)

Is that x is no solution.

So it represents a linear transformation, not a translation

Then you need to add a dimension, which is 4-dimensional

That is, in 4D space, the multiplication still does not represent the translation of 4D, 4 D 0 vector is always changed to 0 vectors, we add a dimension, is to shear 4D space, 3D space plane does not go through the 4D origin, you can use

4D Shear represents 3D panning

The meaning of 4x4:

In computer graphics, coordinate transformations are usually not single, and a geometry may be designed with multiple shifts, rotations, and scales in each frame,

These changes we usually get a final change matrix using the method of connecting each sub-change matrix, thus reducing the computational amount

Does not change the collinear/coplanar of points after affine transformation, and also maintains proportional

If we want to transform a triangle, we only need to transform the three fixed-point v1,v2,v3 t can be, for the original Edge v1v2 on the point, the transformation must still be on the side of T (v1) T (v2).

Affine transformation and homogeneous coordinates