To begin, let us define projection spaces and discuss them to introduce the topic.

1. Projection Spaces

A projection space is defined as the space formed by lines of a certain dimension. For instance, in R2R2, lines passing through the origin are called rays. These rays can be considered as continuous extensions in a two-dimensional plane. For example, if the point (x,y)(x,y) lies on a ray, then the point (−x,−y)(−x,−y) also lies on the same ray, meaning these points define each other. Therefore, since there are no gaps between these rays, the intersection points of lines passing through the origin are not considered.

If we disregard the intersections at the origin, we can remove the origin from the equation and define the rays using a specific equivalence principle. This relationship is defined as follows:

(x, y) ∼ (x*, y*) ⟺ (x*, y*) = λ(x, y) for some λ ∈ ℝ \ {0}

• Here, the symbol ∼ indicates that two points lie on the same ray.
• λλ is a scalar multiplier and a non-zero real number.
• (x,y)(x,y) and (x∗,y∗)(x∗,y∗) represent two points or vectors.

This formula can be interpreted as follows:

• This statement means that the point (x∗,y∗)(x∗,y∗) is a scalar multiple of the point (x,y)(x,y). In other words, the point (x∗,y∗)(x∗,y∗) is obtained by scaling the point (x,y)(x,y) by a certain factor. This implies that both points lie on the same line, as both pass through the origin.

In conclusion, in R2∖{(0,0)}R2∖{(0,0)}, each ray can be defined by two different unit vectors. This ensures that each ray not only excludes the origin but also includes other points on the ray. In this context, the equality of two points means that they lie on the same ray, which is a fundamental property of projection spaces.