Projective Geometry

Projective geometry sits at the top of the Erlangen hierarchy: the geometry of the largest classical transformation group, with the fewest invariants but the greatest unifying power. Its founding move is to adjoin points at infinity so that any two lines meet — abolishing the special case of parallels that generated the entire parallel-postulate saga. Its payoff for this spine is the Cayley–Klein construction: fixing a single conic inside the projective plane, and measuring distance by the cross-ratio to that conic, recovers all three classical geometries — Euclidean, hyperbolic, and elliptic — from one projective source. This is the common ancestor the models and the Erlangen program kept pointing back to.

References: Coxeter, Projective Geometry and The Real Projective Plane; Richter- Gebert, Perspectives on Projective Geometry; Hartshorne, Geometry: Euclid and Beyond, chs. 5–6; Berger, Geometry I, chs. 4–6.

1. The Projective Plane

The real projective plane ${\mathbb{RP}}^2$ is the set of lines through the origin in ${\mathbb{R}}^3$; equivalently, ${\mathbb{R}}^3\setminus \{0\}$ with $\mathbf{x}\sim \lambda \mathbf{x}$ for $\lambda \ne 0$. A point is a homogeneous coordinate $[x:y:z]$, defined up to scale. Concretely,

$${\mathbb{RP}}^2=\underset{[x:y:1]}{\underbrace{{\mathbb{R}}^2}}\cup \underset{\text{line at infinity}}{\underbrace{\{[x:y:0]\}}}.$$

The ordinary affine plane sits inside as $z=1$; the extra points $[x:y:0]$ form the line at infinity, one point for each direction. Two Euclidean parallels — same direction — now meet at their common point at infinity. This is the same antipodal-quotient space $S^2/\{\pm 1\}$ that appears as elliptic geometry, here carrying a different (projective, not metric) structure.

The transformations are the projectivities (homographies): maps induced by invertible linear maps of ${\mathbb{R}}^3$, forming the group $PGL(3,\mathbb{R})=GL(3)/{\mathbb{R}}^{\times }$. They send lines to lines and preserve incidence but not lengths, angles, or parallelism.

2. Duality

The defining symmetry of projective geometry, absent from Euclidean geometry:

Principle of Duality. In the projective plane, every true statement remains true when "point" and "line" are interchanged (and "lies on" ↔ "passes through"). Points and lines are on completely equal footing.

"Two points determine a line" dualizes to "two lines determine a point" — true without exception because parallels have been abolished. Duality halves the theorems one must prove (each comes with a dual for free) and is the cleanest sign that projective geometry is more symmetric than Euclidean geometry. It is powered by the fact that lines in ${\mathbb{RP}}^2$ are themselves parametrized by a projective plane (the dual plane).

3. The Cross-Ratio: the Fundamental Invariant

Projectivities destroy distance and ratio, but they preserve one number built from four collinear points $A,B,C,D$ — the cross-ratio

$$(A,B;C,D)=\frac{\overline{AC}\cdot \overline{BD}}{\overline{BC}\cdot \overline{AD}}.$$

Invariance. The cross-ratio of four collinear points is unchanged by every projectivity. It is the fundamental invariant of projective geometry — every projective quantity is built from it.

The cross-ratio is what survives when length (Euclidean), ratio-of-lengths (affine), and even ratio-along-a-line degrade under the larger projective group — the top of the invariant hierarchy. It is also the key to turning projective geometry back into metric geometry, next.

4. The Cayley–Klein Construction

Here projective geometry repays the whole spine. Fix a conic $\mathcal{C}$ (the absolute) in ${\mathbb{RP}}^2$. Define the distance between two points $P,Q$ by the cross-ratio with the two points $A,B$ where line $PQ$ meets $\mathcal{C}$:

$$d(P,Q)=\frac{c}{2}|\ln (A,B;P,Q)|.$$

Because the cross-ratio is projectively invariant (§3), this distance is preserved by exactly the projectivities that fix $\mathcal{C}$ — turning the subgroup $\mathrm{Stab}(\mathcal{C})\subset PGL$ into a group of isometries. The signature of the absolute conic selects the geometry:

The real-conic case is precisely the Beltrami–Klein model: points inside the circle, lines are chords, and the Cayley–Klein cross-ratio distance is the hyperbolic metric. So:

Projective geometry is the common ancestor. All three constant-curvature geometries arise from a single projective plane by choosing an absolute conic of each signature. This is the exact projective mirror of the Erlangen classification of the three geometries as the conic-preserving subgroups of $PGL$.

5. Classical Theorems

Two theorems display the flavour of pure projective reasoning — both are about incidence alone, with no metric content, and both are self-dual or come in dual pairs:

• Desargues' theorem. If two triangles are perspective from a point (lines joining corresponding vertices concur), they are perspective from a line (intersections of corresponding sides are collinear) — and conversely. Its truth in a projective plane is equivalent to the plane being coordinatizable over a division ring.
• Pappus' theorem. Given two lines with alternating points $A,B,C$ and $A^{\prime },B^{\prime },C^{\prime }$, the three intersection points of the "cross-joins" $A{B}^{\prime }\cap A^{\prime }B$, $B{C}^{\prime }\cap B^{\prime }C$, $C{A}^{\prime }\cap C^{\prime }A$ are collinear. Pappus holds iff the coordinate ring is commutative — projective geometry secretly encoding field axioms.

These illustrate why projective geometry is the natural setting for the incidence substrate beneath all the metric geometries: it is the geometry of pure position, prior to any notion of distance.

6. Place in the Spine

Projective geometry closes the loop opened by Euclid:

• It removes the parallel exception by adjoining points at infinity, dissolving the very distinction the parallel postulate legislated.
• Via the Cayley–Klein absolute (§4) it regenerates all three classical geometries as metric refinements of one projective space — the concrete realization of the Erlangen hierarchy with projective geometry at the top.
• It supplies the Beltrami–Klein model that proved hyperbolic geometry consistent, so the logical vindication of non-Euclidean geometry runs through projective geometry.

From the summit of the hierarchy, the whole spine reads as one structure: a single projective plane, and a menu of absolute conics whose signatures are the curvatures $K>0$, $K=0$, $K<0$ — Euclid's fifth postulate demoted from an axiom to a choice of conic.