Unit 3: Involutions, Planar Polar Systems & Order Curves (Conics)

Karl Georg Christian von Staudt — Geometrie der Lage (1847), §§ 9 & 18–19 (arts. 103–115, 235–257)

[Folio § 9 | p. 49]

§ 9. Involutions on Uniform Fundamental Forms (arts. 103–115)

Art. 103. When a non-identical projectivity of a uniform fundamental form onto itself interchanges even a single pair of distinct elements ($A \mapsto A'$ and $A' \mapsto A$), it interchanges every element with its image ($X \leftrightarrow X'$) and is called an involution (Involution).

Proof. By the Fundamental Theorem (Art. 96), a projectivity is uniquely determined by three pairs of corresponding elements. Consider the given projectivity $\tau$ sending $(A, A', B) \mapsto (A', A, B')$ together with its inverse $\tau^{-1}$, which sends $(A', A, B') \mapsto (A, A', B)$. On the projective carrier, the ordered tetrad $(A, A', B, B')$ is projectively equivalent to $(A', A, B', B)$ via the quadrangle interchange of conjugate pairs (§ 6, Art. 72)! Hence $(A, A', B) \mapsto (A', A, B')$ also carries $B' \mapsto B$, so $\tau = \tau^{-1}$ everywhere. $\blacksquare$

Art. 106 ( Quadrangle Theorem of Desargues–Staudt ). The three pairs of opposite sides of any complete plane quadrangle $EFGH$ are cut by every transversal $\ell$ not passing through a vertex in three pairs of an involution.

Art. 109 (Hyperbolic and Elliptic Involutions). Two distinct pairs $(A, A')$ and $(B, B')$ uniquely determine an involution:

  1. If $(A, A')$ and $(B, B')$ do not separate one another, the involution is hyperbolic and possesses exactly two real double elements (Ordnungselemente $M, N$), which are simultaneously harmonic conjugates with respect to every conjugate pair $(X, X')$ of the involution: $(M, N; X, X') = -1$.

  2. If $(A, A')$ and $(B, B')$ do separate one another, corresponding points traverse the closed carrier in the same cyclical sense and never meet; the involution is elliptic and possesses no real double elements (in 1856 Beiträge Heft 1, von Staudt represents a pair of conjugate imaginary points synthetically as an elliptic involution equipped with a cyclical sense!).

[Folio §§ 18–19 | p. 129]

§§ 18–19. Planar Polar Systems and Order Curves (arts. 235–257)

Art. 235 (§ 18 — Planar Polarity). A reciprocal correspondence (Reciprocität or correlation) in a plane assigns to every point $P$ a unique straight line $p = \pi(P)$ and to every straight line $g$ a unique point $G = \pi(g)$ such that collinear points correspond to concurrent lines and harmonic tetrads of points correspond to harmonic pencils of four rays. When this correspondence is involutory—so that whenever $Q$ lies on the line $\pi(P)$, $P$ likewise lies on the line $\pi(Q)$—it is called a planar polar system (ebenes Polarsystem), $P$ is the pole (Pol) of $p$, and $p$ is the polar (Polare) of $P$.

Art. 238. Three points $A, B, C$, each of which is the pole of the opposite side ($a = [BC], b = [CA], c = [AB]$), form a polar triangle (Polardreieck or self-polar triangle). A planar polar system is uniquely determined by a polar triangle $ABC$ together with one additional pole–polar pair $(P, p)$ in general position.

Art. 242 (§ 19 — Von Staudt’s Intrinsic Definition of Conics). In a planar polar system $\pi$, either no point lies upon its own polar (elliptic polar system), or infinitely many points lie upon their own polars (hyperbolic polar system). In the latter case:

  • The locus of all points $X$ of the plane that lie upon their own polars ($X \in \pi(X)$) is called a curve of the second order or order curve (Ordnungscurve — a point-conic).

  • Dually, the envelope of all straight lines $x$ of the plane that pass through their own poles ($\pi(x) \in x$) is a curve of the second class (Hüllcurve — the tangent-conic of the exact same curve).

Von Staudt §§ 18–19 (Arts. 246–257): Conic as Self-Conjugate Locus of a Polarity & Pascal's Line Curve of the second order defined as P in pi(P) with pole Q, chord polar pi(Q) = [P2 P3], tangents pi(P2), pi(P3), and collinear Pascal points M1, M2, M3. Q (pole of [P₂P₃]) P₁ P₂ ∈ π(P₂) P₃ ∈ π(P₃) P₄ P₅ P₆ M₁ M₂ M₃ Geometrie der Lage §§ 18–19: Conic defined purely as self-conjugate locus P ∈ π(P), yielding tangents [Q P₂], [Q P₃] and Pascal line [M₁ M₂ M₃]
**Von Staudt's Polar Definition of the Conic (§§ 18–19, arts. 235–249):** The conic $\mathcal{C}$ is defined purely as the locus of self-incident points $X \in \pi(X)$. From an exterior pole $P$, the polar $p = [T_1 T_2]$ joins the contact points of the self-conjugate tangents, and every secant through $P$ is cut harmonically: $(T_A, T_B; P, Q) = -1$.

Art. 245. Every straight line $g$ in the plane meets an order curve $\mathcal{C}$ in at most two points (namely the double points, if real, of the involution of conjugate points induced by $\pi$ upon $g$):

  • If the induced involution on $g$ is elliptic, $g$ is an exterior line (Sekantenlose) meeting $\mathcal{C}$ in zero real points.

  • If $g$ lies on its own pole $T = \pi(g)$, the involution degenerates at $T$: $g$ meets $\mathcal{C}$ in the single contact point $T$ and is the tangent to $\mathcal{C}$ at $T$.

  • If the induced involution on $g$ is hyperbolic with double points $T_A, T_B \in \mathcal{C}$, $g$ is a secant cutting $\mathcal{C}$ in $T_A$ and $T_B$, and every pair of conjugate points $(P, Q)$ on $g$ divides $(T_A, T_B)$ harmonically: $(T_A, T_B; P, Q) = -1$.

Art. 253 (Steiner’s Generation & Pascal–Brianchon Duality as Corollaries). Because the pencil of rays joining a variable point $X \in \mathcal{C}$ to two fixed points $A, B \in \mathcal{C}$ corresponds projectively to the range of intersections of the polar $\pi(X)$ (the tangent at $X$) with the fixed tangents at $A$ and $B$, Steiner’s projective pencil generation of conics, Pascal’s inscribed Hexagon Theorem, and Brianchon’s circumscribed Hexagon Theorem follow immediately without a single cone in three-space or quadratic equation!