I am reading through the first chapter of Grunbaum's book Convex Polytopes, in which he gives the following definition of projective transformations $\mathbb{R}^d\to\mathbb{R}^d$:
$$Tx = \frac{Ax + b}{\langle c, x \rangle + \delta}$$
Where $T$ is the projective transformation, $A$ is a $d$-by-$d$ matrix, $b, c$ are vectors in $\mathbb{R}^d$ and $\delta$ is a real number. The claim is, given that $\delta, c$ are both nonzero, such a map sends colinear points to colinear points. I've done the computations and have verified this, but I feel that I'm still missing an intuitive understanding of what exactly such a transformation looks like.
I can see that in the numerator we have an affine transformation, but I am puzzled by the denominator, specifically the meanings of the parameters $c$ and $\delta$.
Thanks
$d$-dimensional real projective space, denoted $\mathbb{R}P^d$, is the set of all lines in $\mathbb{R}^{d+1}$ passing through the origin. Given a nonzero $x = (x^0,\dots, x^d) \in \mathbb{R}^{d+1}$, let $[x] = [x^0, x^1, \dots, x^d] \in \mathbb{R}P^d$ denote the line containing $(x^0,\dots, x^d)$.
Any invertible linear transformation $A: \mathbb{R}^{d+1} \rightarrow \mathbb{R}^{d+1}$ induces a map $\hat{A}: \mathbb{R}P^d \rightarrow \mathbb{R}P^d$, where $$ \hat{A}([x]) = [Ax]. $$ Such maps are called projective transformations.
There is an coordinate map \begin{align*} \Phi: \mathbb{R}^d &\rightarrow \mathbb{R}P^d\\ (x^1, \dots, x^d) &\mapsto [1, x^1, \dots, x^d]. \end{align*} These coordinates are called affine coordinates. The image of this map is $$ \mathbb{A} = \{ [x^0, \dots, x^d]\ :\ x^0 \ne 0 \}. $$ The inverse map is \begin{align*} \Phi^{-1}: \mathbb A &\rightarrow \mathbb{R}^d\\ [x^0, \dots, x^d] &\mapsto \left[1, \frac{x^1}{x^0}, \dots, \frac{x^d}{x^0}\right]. \end{align*}
By restricting the projective transformation $\hat{A}$ to $\mathbb{A}$, you get a map $$ \Phi^{-1}\circ \hat{A}\circ \Phi: \mathbb{R}^d\backslash H \rightarrow \mathbb{R}^d, $$ where $H$ is an affine hyperplane. If you work out the formula for this map, you will see that it matches your formula for $T$, where $$ H = \{ \langle c,x\rangle + \delta = 0\}. $$