The one annoying exception
All the way up this ladder, one fact has quietly cost you. Two distinct points always determine exactly one line — clean, no exceptions. But the mirror statement, two distinct lines always meet in exactly one point, has a gaping hole in it: parallel lines never meet at all. You met this asymmetry first as Euclid's troublesome parallel postulate, and again as Playfair's axiom, the promise that through a point off a line there is exactly one parallel. Parallels are the reason 'two lines meet in a point' has to keep saying 'unless they happen to be parallel'.
That little 'unless' is more than an inconvenience — it is a crack running through the whole subject. Every theorem about intersecting lines has to fork into a generic case and a parallel case; every formula about slopes has to guard against the vertical line whose slope is undefined. The dream of projective geometry is to remove the exception entirely, so that the two statements become perfect twins: any two points lie on exactly one line, and any two lines meet in exactly one point, with no escape clause. The whole question is whether we can buy that symmetry honestly, without lying about what a line is.
Adding one point per direction
Here is the move. Take a family of parallel lines — say all the lines pointing northeast, every line with slope 1. They share a single common direction. We invent one new point, call it an ideal point or a point at infinity, and declare that every line in that family passes through it. A different direction — say all the horizontal lines, slope 0 — gets its own, different point at infinity. So there is exactly one new point for each direction, one per family of parallels, no more and no fewer.
Notice the bookkeeping is delicate but exact. A direction is not the same as a slope number alone, but it is the same as a slope together with the choice of going one way along the family — and crucially, a line and its exact opposite heading (slope 1 going up-right, slope 1 going down-left) belong to the same family of parallels, so they share one ideal point, not two. The line itself does not have two ends meeting two different infinities; it has a single point at infinity that both of its far ends run toward. That is the subtle correction beginners often miss.
With these new points in hand the exception dissolves. Two lines that used to be parallel now genuinely do share a point — their common ideal point — so they meet exactly once, just like any other pair. Two lines that already crossed at an ordinary point still cross there and nowhere else, because they point in different directions and so head toward different points at infinity. Either way, exactly one meeting point. The ordinary plane plus all its ideal points is the projective plane, and its incidence rule is finally free of buts: two points, one line; two lines, one point.
All the new points lie on one new line
We just added a whole crowd of new points — one for every possible direction. To keep the rules consistent, we must decide how lines treat them. The natural and forced choice: collect all the points at infinity into a single new line, the line at infinity. So now the projective plane has ordinary points, ideal points, ordinary lines (each carrying exactly one ideal point), and one extra line — the line at infinity — that carries every ideal point and no ordinary one.
Check that this choice keeps the twin laws intact. Two ordinary lines that cross at a finite point share that finite point and no other; two ordinary parallels share their common ideal point and no other; and any ordinary line meets the line at infinity in exactly its one ideal point. Every pair of distinct lines meets exactly once — even when one of them is the new line. The construction is not ad hoc decoration; it is the unique way to add points so the 'two lines, one point' law holds with no exceptions at all.
Why the painter saw it first
If this feels abstract, look at any photograph of a long straight road or railway track. The two rails are perfectly parallel, yet in the picture they visibly converge and meet at a point on the horizon — the vanishing point. That is not an illusion of bad measurement; it is exactly a point at infinity made visible. Renaissance painters discovering perspective were, without the vocabulary, drawing the projective plane: every family of parallel lines in the scene races to its own single vanishing point, and all those vanishing points sit along one horizon line — the line at infinity of the picture.
This is also why the right gadget for projective geometry is projection itself. Imagine your eye at a point, looking at a flat tabletop, and casting its image onto a tilted canvas. Lines stay lines, but lengths, angles, and parallelism are all scrambled — a circle on the table can show up as an ellipse on the canvas, and the table's faraway edge can land as an ordinary line on the canvas. What survives such a projection is precisely what projective geometry studies. In the language of the Erlangen program you met earlier, projective geometry is the geometry whose allowed motions are these projections, and its theorems are the facts they leave unchanged.
What you keep, what you let go
Be clear-eyed about the trade. To win the exception-free incidence rule, projective geometry gives up a lot. Distance is gone — there is no '|AB|' between an ordinary point and a point at infinity. Angle is gone, so 'perpendicular' has no meaning. Even parallel loses its job, because lines that used to be parallel now meet like everyone else; parallelism was a feature of affine geometry, the gentler relaxation that keeps parallels but drops distance and angle. Projective geometry sits one rung looser still: it keeps only what mere straightness and incidence can express.
So what is left to do mathematics with? Plenty, and it is sturdier for being so spare. Whether three points are collinear, whether three lines pass through a common point, whether a point lies on a given line — all of this survives every projection untouched, because projection sends lines to lines and preserves who-lies-on-what. The deep theorems of the next guides live here: Desargues' theorem about two triangles in perspective, Pappus' theorem about six points on two lines, and a single number — the cross-ratio — that projection cannot disturb. Drop distance and angle, and these incidence facts stand out in stark relief.
WHAT EACH GEOMETRY KEEPS (looser as you go down) Euclidean : distance, angle, parallel, straightness, incidence Similarity : ratio, angle, parallel, straightness, incidence Affine : - parallel, straightness, incidence Projective : - - straightness, incidence parallels meet? Euclid/affine: NO projective: YES (at infinity)
Where this rung is heading
You now hold the founding idea: adjoin one ideal point per direction and one line at infinity, and incidence becomes flawlessly symmetric. But so far the points at infinity are described in words — 'the point where all slope-1 lines meet' — which is awkward to compute with. The very next guide fixes that by giving every point, ordinary or ideal, an honest triple of numbers through homogeneous coordinates, so the point at infinity gets coordinates as concrete as any other point, and a line becomes an equation you can solve.
From there the rung's real payoff opens up. Because points and lines now obey perfectly mirrored laws, every theorem comes with a twin you get by swapping the words 'point' and 'line' — that is the principle of duality, guide 3. Guide 4 mines the incidence-only world for its classical gems, Desargues, Pappus, and the projection-proof cross-ratio. And guide 5 returns to the conics you studied as plane sections, only to find that in the projective plane the ellipse, parabola, and hyperbola stop being three things and become one, distinguished merely by how they meet the line at infinity.
Keep one honest caveat in your pocket as you climb. None of this says Euclid was wrong, any more than the discovery of non-Euclidean geometry did. Projective geometry is not a correction of the plane you knew; it is a different, deliberately looser game with its own rules, and the Euclidean plane sits happily inside it as the part you see once you pick out a line to call 'infinity'. You are gaining a wider stage, not trading your old one away.