PERSPECTIVE SPACE

In a $R_3$ euclidean space such as the 3D world there is no way to represent a point to infinity and parallel lines does not intersect or intersect to infinity.

To address this the perspective space $P_3$ is created, inside this space each $R_3$ point is mapped to a class of points defined as follows:

$$(x,y,z)=(x,y,z,k)\forall k\in \ne 0$$

POINT TO INFINITY REPRESENTATION

In this space points to infinity can be represented, so given a line equation like this:

$$\left[\begin{array}{c}{x}_0\\ y_0\\ z_0\end{array}\right]+\lambda \times \left[\begin{array}{c}a\\ b\\ c\end{array}\right]=\left[\begin{array}{c}{x}_0+\lambda a\\ y_0+\lambda b\\ y_0+\lambda c\end{array}\right]$$

we can represent a point to infinity by the limit $\lambda \to \infty$ as follows (here it’s used the value $1$ for the $k$ coordinate):

$$li{m}_{\lambda \to \infty }\left[\begin{array}{c}{x}_0\\ y_0\\ z_0\\ 1\end{array}\right]+\left[\begin{array}{c}\lambda a\\ \lambda b\\ \lambda c\\ 1\end{array}\right]=\left[\begin{array}{c}\frac{x_0}{\lambda }+a\\ \frac{y_0}{\lambda }+b\\ \frac{y_0}{\lambda }+c\\ \frac{1}{\lambda }\end{array}\right]=\left[\begin{array}{c}a\\ b\\ c\\ 0\end{array}\right]$$

The class $(x,y,z,0)$ represents all of the possible points to infinity. This representation it’s not possible in the $R_3$

DEFINITION EXTENSION

The perspective space definition can be extended to a $N$-dimensional space $R_n\to P_n$

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