Unit 3 Summary: Involutions, Polarities & Conics Without Cones

Analytical Synopsis of Geometrie der Lage (1847), §§ 9 & 18–19

1. Single-Swap Criterion for Involution (§ 9, art. 103): Any projectivity on a 1D carrier that swaps a single distinct pair $A \leftrightarrow A'$ swaps every point with its image ($\tau^2 = \mathrm{id}$).

2. Hyperbolic vs. Elliptic Involutions (§ 9, art. 109): Non-separating pairs determine two real double points dividing every conjugate pair harmonically; separating pairs yield an elliptic involution without real double points.

3. Planar Polar System (§ 18, arts. 235–238): An involutory point-line correlation $\pi$ ($Q \in \pi(P) \iff P \in \pi(Q)$) is determined by a self-polar triangle plus one generic pole–polar pair.

4. Intrinsic Conic Definition (§ 19, arts. 242–245): Defines an order curve (conic) purely in the plane as $\{X : X \in \pi(X)\}$, instantly unifying tangents, harmonic pole-polar secants, Steiner’s pencil generation, and Pascal–Brianchon duality.