1. Translation in Homogeneous Coordinates
Within a frame, each affine transformation is represented by a 4x4 matrix of the form:

2. A point p will be moved to point p’ by a displacement distance d: p’ = p + d. The translation matrix is defined as T (dx ,dy,, dz):
The inverse of a translation matrix is obtained by applying an inversion algorithm or by noting that, if we displace a vector by d we can return it to its original position by
Displacing it by –d: T-1(dx, dy, dz)= T(-dx ,-dy,, -dz)

3. Size Scaling in Homogeneous Coordinates
For both the size scaling and rotation, there is a fixed point that is unchanged by the transformation.
These three can be combined as: p’ = Sp

4. The inverse of size scaling matrix is:

5. Example:
Size scale the p(2, -3, 5) by half and move it 3 unit distance away on x and -1 distance away on y.

6. Rotation in Homogeneous Coordinates

8. The rotation can be reversed for all of them using the following procedure:

9. Example:
Rotate p(2, -3, 5) by 45 around x axis.

10. The inverse can be computed as:
Shear in Homogeneous Coordinates
Shear in x direction
The inverse can be computed as: