These parallel lines appear to intersect in the vanishing point "at infinity". In a projective plane this is actually true.
In mathematics, a projective plane is a geometric structure that extends the concept of a plane. In the ordinary Euclidean plane, two lines typically intersect in a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus any two distinct lines in a projective plane intersect in one and only one point.
A projective plane is a 2-dimensional projective space, but not all projective planes can be embedded in 3-dimensional projective spaces. Such embeddability is a consequence of a property known as Desargues' theorem, not shared by all projective planes.
A projective plane consists of a set of lines, a set of points, and a relation between points and lines called incidence, having the following properties:
Given any two distinct points, there is exactly one line incident with both of them.
Given any two distinct lines, there is exactly one point incident with both of them.
There are four points such that no line is incident with more than two of them.
The second condition means that there are no parallel lines. The last condition excludes the so-called degenerate cases (see below). The term "incidence" is used to emphasize the symmetric nature of the relationship between points and lines. Thus the expression "point P is incident with line ℓ " is used instead of either "P is on ℓ " or "ℓ passes through P ".