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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

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    Think of it as perpsective projection onto the hyperplane consisting of points with last coordinate equal to $1$. It maps all points along a line through the origin (minus the origin itself) onto the point of intersection of this line with the hyperplane. – 2017-02-07
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    can u please elaborate the point (It maps all points along a line through the origin (minus the origin itself) onto the point of intersection of this line with the hyperplane.) ? – 2017-02-07
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    The parameterization $t(z,1)$ for some fixed $z$ describes a line through the origin. $P$ maps every point on this line to $z$. Try a few examples in $\mathbb R^3$ to visualize what’s going on. – 2017-02-07
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    A related topic is the simple pinhole camera model that's used in computer graphics and computer vision. Perspective projection is a special case where the eye of the camera is at the origin and the camera is looking along the the last coordinate axis. – 2018-07-06

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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.

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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".