There are different ways to think about geometry in general and projective geometry in particular. The axiomatic approach requires no philosophical definition of what a point or a line actually “is”, just a list of properties (axioms) that they satisfy. The theorems of geometry are all statements that can be deduced from these properties. Euclid wrote down a list of these axioms – five of them are called the postulates.

The first four postulates are self-evident, and say things like “for any pair of points, there is a unique line passing through both of them”. However, the fifth postulate was quite complicated, which postulates that, for any line L and a point P not on L, there exists a unique line that is parallel to L (never meets L) and passes through P. For this reason, the fifth postulate is called the “Parallel Postulate”. It is possible to assign meanings to the terms “Point” and “Line” in such a way that they satisfy the first four postulates but not the parallel postulate.

These are called “non-Euclidean Geometrics”. Projective geometry is not really a typical non-Euclidean geometry, but it can still be treated as such. In this axiomatic approach, projective geometry means any collection of things called “points” and things called “lines” that they obey the same first four basic properties that points and lines in a familiar flat plane do, but which, instead of the parallel postulate, satisfy the following opposite property instead: “Any two lines intersect (in exactly one point). This means that there is no such thing as a pair of parallel, non-intersecting lines in projective geometry. In projective geometry, the words: points and lines are interchangeable. The basis axiom that “for any two points, there is a unique line that intersects both these points” when turned around, becomes “for any two lines, there is a unique point that intersects (i.e. lies on) both those lines”. There is a complete duality between points and lines in projective geometry.

However, this says nothing about whether such concepts would be interesting, relevant and or have any relation whatsoever to the normal concepts of lines and planes in Euclidean geometry.

Projective geometry can also be thought of as the collection of all lines through the origin in a three-dimensional space. That is, each point of projective geometry is actually a line through the origin in three-dimensional space. The distance between 2 points can be thought of as the angle between the corresponding lines. A line in projective geometry is really a family of lines through the origin in three-dimensional space. There is yet another way to understand projective geometry: it is the geometry of curves on a rather weird surface. This surface cannot be embedded in the standard 3-dimensional world we are living in.

We need to live in four dimensions to visualize it completely. Take a sphere, (in mathematics, spheres are hollow surfaces and not solid balls) and think of gluing together all pairs of antipodal points (antipodal points are those points which are on the most opposite part of the sphere from each other. E.g.: if the center of the sphere is at the origin, the antipodal point of (x, y z) is (-x, -y, -z) so that they become the same point. Another way to think of it is to take the top hemisphere, and then “seal it up” into a closed surface by gluing each point on the equator to its opposite point. Now, if you put a sphere with its center at the origin in three-dimensional space, then every line through the origin passes through exactly two antipodal points in the sphere (and therefore exactly one point in our surface, after those 2 antipodal points get glued into the same spot). Therefore, there is one-to-one correspondence between points in projective space and points on this weird surface.

When the antipodal points are glued together, the postulate “ any two lines intersect in exactly one point” holds true. Therefore, a “straight line” on a surface like a sphere is this glued surface is not really straight but still can be defined as giving the shortest distance between two points.

On a sphere, the shortest path between 2 points is an arc that is part of a “great circle” on the sphere (whose center is the center of the sphere) and so, the “straight lines” on a sphere are the great circles. All these great circles intersect only at the equator. Suppose you set up two perpendicular lines at two points on a line on the sphere, they will not form parallel lines like on a plane surface, but intersect each other. In summary, as per usual geometry, parallel lines do not meet (and infinity is not defined) and in projective geometry, any two lines intersect in exactly one point. For this reason, when the infinite point is present in a particular geometry, no one talks of parallel lines at all.