Counting for rigidity under projective transformations in the plane

TL;DR

This work studies incidence geometries in the real projective plane through a linearized incidence constraint framework. It introduces the projective rigidity matrix $M(S,\mathbf{p},\mathbf{l})$, defines infinitesimal motions as its kernel, and analyzes rigidity vs. infinitesimal and second-order rigidity, including pinning strategies. The paper shows that infinitesimal (and second-order) rigidity imply rigidity, but rigidity does not imply infinitesimal rigidity, and it develops the notion of projective self-stresses via the row-co-kernel, revealing a three-fold balance that interprets classic incidences theorems such as Desargues, Pappus, and Pascal as constraints or consequences in this framework. It also explores alternate realizations and the depth of constraint independence in fundamental configurations, offering several open problems and directions for generalization and higher-dimensional extensions. The results provide a projective-geometric analogue of rigidity theory, connecting algebraic, combinatorial, and geometric aspects of incidence structures with potential implications for geometric theorem proving and constraint systems.

Abstract

Let $P$ be a set of points and $L$ a set of lines in the (extended) Euclidean plane, and $I \subseteq P\times L$, where $i =(p,l) \in I$ means that point $p$ and line $l$ are incident. The incidences can be interpreted as quadratic constraints on the homogeneous coordinates of the points and lines. We study the space of incidence preserving motions of the given incidence structure by linearizing the system of quadratic equations. The Jacobian of the quadratic system, our projective rigidity matrix, leads to the notion of independence/dependence of incidences. Column dependencies correspond to infinitesimal motions. Row dependencies or self-stresses allow for new interpretations of classical geometric incidence theorems. We show that self-stresses are characterized by a 3-fold balance. As expected, infinitesimal (first order) projective rigidity as well as second order projective rigidity imply projective rigidity but not conversely. Several open problems and possible generalizations are indicated.

Figures (20)

• Figure 1: a) Complete quadrangle b) Complete quadrilateral
• Figure 2: A movable configuration. (The animation controls work depending on one's pdf viewer.)
• Figure 3: The construction of a Desargues Configuration. a) an initial isostatic $K_4$: $a,b,c,d$ pinned. (b) Adding three points (red) on three of the edges to add a second triangle $a',b', c'$ and c) constructing the line of perspective of the two triangles from three pairs of edges where the collinearity of the points is a constraint guaranteed by Desargues Theorem -- a constraint which is dependent on the other constraints.
• Figure 4: a) A hexagon with a Pascal line. b) Animation with $A$, $B$, $A'$, $B'$, pinned. $X$, $Y$, free to move subject to the collinearity $[[[AB][A'B']],X,Y]=0$. Points $C = [[BX][A'Y]]$, $C' = [[AY][B'X]]$ determined.
• Figure 5: a) An infinitesimal $x$-dilation, b) An infinitesimal $x$-shear

Theorems & Definitions (29)

• Theorem 1.1: Theorem of Desargues
• Theorem 1.2: Theorem of Pappus
• Remark 1
• Example 4.1
• Theorem 5.1
• proof
• Theorem 5.2
• proof
• Theorem 6.1
• proof