a projective conic

In ordinary geometry the ellipse, parabola, and hyperbola look like three different curves — one closed, one open with a single branch, one with two branches running off to infinity. Projective geometry delivers a startling unification: in the projective plane these are all the SAME kind of object, a single shape called a conic. The apparent differences are just an accident of where each curve happens to cross the line at infinity, which a projective transformation can move at will.

There are two equivalent ways to pin a projective conic down. The algebraic way: a conic is the set of points (x : y : z) satisfying one homogeneous second-degree equation a x^2 + b y^2 + c z^2 + d x y + e x z + f y z = 0. The familiar trio falls out by intersecting with the line at infinity z = 0 — an ellipse misses it (no real infinity-points), a parabola touches it (a double point, tangent there), a hyperbola crosses it twice (its two asymptotic directions). The synthetic way, due to Jakob Steiner, uses no coordinates at all: take two distinct points and a projectivity (a cross-ratio-preserving correspondence) between the pencils of lines through them; the points where corresponding lines meet trace out exactly a conic. This 'projective generation' is the deep reason a conic, like a line, is governed entirely by incidence and cross-ratio.

The projective viewpoint matters because it makes one theory do the work of three and exposes the symmetries the metric picture hides — pole-polar duality, the conic through five points, Pascal's theorem all live here. Two honest cautions. First, projectively there is only one kind of non-degenerate conic; 'ellipse versus hyperbola' is a metric or affine distinction, recovered only after you single out a line at infinity, not a projective one. Second, a conic can degenerate — if its defining quadratic factors, the 'conic' collapses into a pair of lines (or a single repeated line); these degenerate cases are exactly where Pascal's theorem turns into Pappus' theorem.