what is meant by perspective function?

Question

the perspective function $P : \mathbb{R}^{n+1} \to \mathbb{R}^n$, with domain dom $P = \mathbb{R}^n \times \mathbb{R}^{++}$, as $P(z,t) = \frac{z}t$. (Here $\mathbb{R}^{++}$ denotes the set of positive numbers: $\mathbb{R}^{++} = \{x \in \mathbb{R} | x > 0\}$.) The perspective function scales or normalizes vectors so the last component is one, and then drops the last component.

I was reading convex optimisation and I am not able to understand what is a perspective function ! can any one explain please

Answer 1

For any fixed $z$, $P$ maps all points of the form $t(z,1)$, $(t>0)$ to $z$. These points form a ray emanating from the origin, so you can think of $P$ as the perspective projection onto the the hyperplane that consists of all points with last coordinate equal to $1$. The image of a point in $\mathbb R^{n+1}$ is then the point of intersection of the ray through the point and this hyperplane.

Answer 2

The perspective function $P:R^{n+1} \to R^n$ is defined mathematically as:

$P(x,t) = x/t, \ \ \ domain\ of\ P = \{ (x,t) | t>0\}$

Intuitively it can be explained as Professor Stephen Boyd explains, "Divide first $n$ elements of the vector by its last component".