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

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)

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

The inverse of size scaling matrix is:

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.

Rotation in Homogeneous Coordinates

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

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

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