In order to help me understand perspective functions, can someone provide me with a simple example that uses numbers?
Let's say I have the perspective function $P : \mathbb{R}^{n+1} \to \mathbb{R}^n$, with domain $P = \mathbb{R}^n \times \mathbb{R}_{++}$, as $P(z,t) = \frac{z}t$. (Here $\mathbb{R}^{++}$ denotes the set of positive numbers: $\mathbb{R}_{++} = \{x \in \mathbb{R} | x > 0\}$.)
Now let's say I have a poyhedron defined in $\mathbb{R}^{3}$ as $\{x \in \mathbb{R}^{3}| -1\leq x\leq1\}$ so it'd be a cube centered at the origin. If I put that set into my prospective function, what comes out? If I go according to the definition of a perspective function, I would imagine the answer would be in $\mathbb{R}^{2}$ with a square at the origin.
But I don't see the math, so how would I describe my cube in order to pass it through a perspective function to get a square?
As you can see, I'm extremely confused, so any help is appreciated