Discover the natural home of algebraic geometry where parallel lines meet and every pair of curves intersects the "right" number of times.
In the ordinary (affine) plane, parallel lines never meet. But in projective space, every pair of lines intersects — parallel lines meet at a point at infinity.
This might sound like a trick, but it's the natural setting for algebraic geometry. In projective space, Bezout's theorem works perfectly: curves of degrees m and n always intersect in exactly m × n points. No exceptions!
Projective space uses homogeneous coordinates [X : Y : Z] instead of (x, y). The key rule: [X : Y : Z] = [λX : λY : λZ] for any λ ≠ 0. Drag the point to explore.
All points [λX : λY : λZ] for λ ≠ 0 represent the same projective point
In projective coordinates, we use three numbers [X : Y : Z] instead of two. The crucial rule: [X : Y : Z] = [λX : λY : λZ] for any λ ≠ 0. To recover affine coordinates, divide by Z: (x, y) = (X/Z, Y/Z).
Watch the magic happen: parallel lines actually meet in projective space! Every set of parallel lines shares a common point at infinity.
In the affine plane, parallel lines never meet. They have the same slope but different y-intercepts.
In projective space, any two distinct lines meet at exactly one point. For parallel lines, that point is "at infinity" — represented by homogeneous coordinates with Z = 0. This is why Bezout's theorem always works in projective space!
Every affine curve extends to a projective curve. The points at infinitytell us about the curve's behavior "at the edges" — like where a parabola "closes up."
A parabola has one point at infinity, where it "closes up".
To get the projective equation, replace x with X/Z and y with Y/Z, then clear denominators. Points at infinity are found by setting Z = 0. A degree-d curve has exactly d points at infinity (counted with multiplicity, over ℂ).
Remember those "missing" intersections? In projective space over ℂ, they're all accounted for. Bezout's theorem works exactly: deg(C₁) × deg(C₂) intersections, every time.
In projective space over complex numbers, curves of degrees m and n intersect in exactly m × n points(counted with multiplicity). No more "missing" intersections!
You've discovered the natural home of algebraic geometry:
Projective space is where algebraic geometry truly lives. Every theorem becomes cleaner, every count becomes exact, and parallel lines finally get to meet!