Lines through the origin, points at infinity, and the projective plane
Projective space RPn (or CPn over the complex numbers) is the space of lines through the origin in kn+1. A point in RP2 is an equivalence class [X : Y : Z] where (X, Y, Z) ≠ (0, 0, 0) and [X : Y : Z] = [λX : λY : λZ] for any nonzero scalar λ. This "extra" dimension eliminates the special cases that plague affine geometry.
In projective space, every two distinct lines meet in exactly one point — there are no parallel lines. The "points at infinity" where parallel lines meet are not special; they are ordinary points of RP2. Projective space is compact, and projective varieties have cleaner intersection theory, which is why algebraic geometers prefer to work projectively.
Each line through the origin represents a point of RP². The translucent plane at z=1 is the standard affine chart — where a line pierces it, you read off an affine coordinate. The red line lies in z=0 and corresponds to a "point at infinity."
Each colored line through the origin represents a point of RP². Where a line pierces the translucent z=1 plane, it gives an affine coordinate. The red line lies in z=0 and never meets the plane — it is a "point at infinity."
A point [X : Y : Z] in RP2 can be "de-homogenized" by setting one coordinate to 1. When Z ≠ 0, we get the affine point (X/Z, Y/Z) in the standard affine chart UZ. Similarly, UX and UYcover the points where X ≠ 0 and Y ≠ 0, respectively. Together, these three charts cover all of RP2.
Points with Z = 0 are the points at infinity — they form a copy of RP1 called the line at infinity. To "projectivize" an affine curve f(x, y) = 0 of degree d, replace x by X/Z and y by Y/Z, then multiply through by Zd to get a homogeneous polynomial F(X, Y, Z). The projective curve V(F) contains the original affine curve plus finitely many points at infinity where the highest-degree part of f vanishes.
In affine space, two lines might be parallel (no intersection), and intersection counts depend on how curves "go off to infinity." Projective space eliminates these edge cases. Bézout's theorem — that two curves of degrees d and e meet in exactly de points (counted with multiplicity) — holds cleanly only in projective space over an algebraically closed field.
Projective space is also compact (in both the Zariski and classical topologies), which means projective varieties have finiteness properties that affine varieties lack. For example, the image of a projective variety under a regular map is always closed — a fact that fails dramatically for affine varieties.