TITLE: What is the difference between affine and projective transformations?
QUESTION [15 upvotes]: I'm trying to grasp the difference between the affine and projective transformations.
I got the point of the "line at infinity", but their matrix representation is not yet clear enough.
Here's the affine transformation $A$
$$
A = 
\begin{bmatrix}
a_1 & a_2 & t_x \\
a_3 & a_4 & t_y \\ 
0  & 0  & 1 
\end{bmatrix}
$$
The matrix $A$ has 2 rotational, 2 scale and 2 translational parameters. 
For projective one (homography) $H$:
$$
H = 
\begin{bmatrix}
h_1 & h_2 & h_3  \\
h_4 & h_5 & h_6 \\ 
h_7 & h_8 & h_9  
\end{bmatrix}
$$
We still have the same parameters as the affine, but we added the last row, which is the projection vector. 
Why do we need to add this vector to the matrix? What does it mean? What is the impact of that vector on the projection itself?

REPLY [3 votes]: The whole point of the representation you're using for affine transformations is that you're viewing it as a subset of projective space. A line has been chosen at infinity, and the affine transformations are those projective transformations fixing this line.
Therefore, abstractly, the use of the extra parameters is to describe where the line at infinity moves during the projective transformation.