A Deeper Understanding / Interpretation of Homographies I currently understand that a homography matrix, which allows for a mapping between planes in 3-dimensions, is a $3\times3$ matrix of the following general form:
$$\begin{bmatrix}
\vert & \vert & \vert \\
r_1 & r_2 & t \\
\vert & \vert & \vert
\end{bmatrix}$$
In my class, we were introduced to the idea that a 2d homography has special cases including similarity transformations, affine transformations, and projective transformations.  
Furthermore, the components of the $H$ matrix were broken up into four "quadrants".
$$H=\left[\begin{array}{c|c}
\\[-1ex]
\quad A\quad & t \\[-1ex]\\\hline
V & v
\end{array}\right]$$
Would someone be able to relate these transformations to the various elements in $H$?
For a similarity matrix, my understanding is that the image can only be rotated in the $xy$ plane, scaled, and translated.  Thus a similarity $H$ matrix would have zeros for elements $h_{31}$ and $h_{32}$ and $1$ for $h_{33}$.
I am unclear how to interpret affine and projective transformations in this way, especially because their definitions are not as clear to me.
 A: 
projective transformations

Any matrix, usually excluding singular matrices (determinant zero). That's the most general case.

affine transformations

Preserve points at infinity, i.e. result has $z=0$ if and only if input has $z=0$. That is what you describe with $h_{31}=h_{32}=0$ and $h_{33}=1$, except that last coordinate $h_{33}$ can in fact be any non-zero value since a multiple of that matrix describes the same transformation. So you'd have $V=0$ and $v\neq0$.
It makes sense to view an affine transformation algebraically, on inhomogeneous coordinates:
$$x'=(h_{11}x + h_{12}y + h_{13})/h_{33}\\
y'=(h_{21}x + h_{22}y + h_{23})/h_{33}$$
Here you can see that $v$ serves as a global denominator, $t$ describes a translation and $A$ a linear 2d transformation.

similarity transformations

These are affine transformations which also preserve angles. For a rotation, the upper left $2\times2$ block $A$ would be a rotation matrix like $\begin{bmatrix}\cos\alpha&-\sin\alpha\\\sin\alpha&\cos\alpha\end{bmatrix}$. But for a similarity these may be scaled as well, so you'd get something of the form $A=\begin{bmatrix}a&-b\\b&a\end{bmatrix}$, together with $V=0$ and $v\neq 0$.
