Title: Counting for rigidity under projective transformations in the plane Show Chinese title

Title: 平面射影变换下的刚性计数

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. Show Chinese abstract

Abstract: 设$P$为点的集合，$L$为欧几里得平面（扩展）中的线的集合，而$I \subseteq P\times L$，其中$i =(p,l) \in I$表示点$p$和线$l$是相关的。 这些关联可以被解释为点和线的齐次坐标上的二次约束。 我们通过线性化二次方程组来研究给定关联结构的保持关联的运动空间。 二次系统的雅可比矩阵，我们的射影刚度矩阵，导致了关联独立性/依赖性的概念。 列依赖性对应于无穷小运动。 行依赖性或自应力允许对经典几何关联定理的新解释。 我们证明自应力由三重平衡所表征。 如预期的那样，无穷小（一阶）射影刚度以及二阶射影刚度都暗示射影刚度，但反之则不成立。 指出了几个开放问题和可能的推广。