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:

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

Point to infinity representation

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

$$ \begin{bmatrix} x_{0} \ y_{0} \ z_{0} \end{bmatrix} + \lambda \times \begin{bmatrix} a \ b \ c \end{bmatrix} = \begin{bmatrix} x_{0} + \lambda a \ y_{0} + \lambda b \ y_{0} + \lambda c \end{bmatrix} $$

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):

$$ lim_{\lambda \to \infty} \begin{bmatrix} x_{0} \ y_{0} \ z_{0} \ 1 \end{bmatrix} + \begin{bmatrix} \lambda a \ \lambda b \ \lambda c \ 1 \end{bmatrix} = \begin{bmatrix} \frac{x_{0}}{\lambda} + a \ \frac{y_{0}}{\lambda} + b \ \frac{y_{0}}{\lambda} + c \ \frac{1}{\lambda} \end{bmatrix} = \begin{bmatrix} a \ b \ c \ 0 \end{bmatrix} $$

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 $R_{n}\rightarrow P_{n}$