Counting for rigidity under projective transformations in the plane

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.