🤖 AI Summary
This paper addresses the oriented affine rigidity of point configurations: if two finite point sets $P$ and $Q$ induce chirotopes with identical realizability across all finite generic extensions, must $P$ and $Q$ be orientation-preserving affinely equivalent?
Method: The authors establish a quantitative link between chirotope extension realizability and geometric rigidity, leveraging tools from oriented matroid theory, real algebraic geometry, genericity analysis, and affine approximation.
Contribution/Results: They prove that this combinatorial condition precisely characterizes orientation-preserving affine equivalence classes of point configurations. Furthermore, they formulate and prove an $varepsilon$-rigidity theorem: for any $varepsilon > 0$, there exists a finite generic extension $widehat{P}$ such that any configuration $Q$ realizing the chirotope of $widehat{P}$ admits an orientation-preserving affine transformation mapping each point of $Q$ into the $varepsilon$-neighborhood of the corresponding point in $P$. This result provides a robust, quantitative foundation for combinatorial rigidity in affine geometry.
📝 Abstract
Let $P$ be a finite full-dimensional point configuration in $mathbb{R}^d$. We show that if a point configuration $Q$ has the property that all finite chirotopes realizable by adding (generic) points to $P$ are also realizable by adding points to $Q$, then $P$ and $Q$ are equal up to a direct affine transform. We also show that for any point configuration $P$ and any $varepsilon>0$, there is a finite, (generic) extension $widehat P$ of $P$ with the following property: if another realization $Q$ of the chirotope of $P$ can be extended so as to realize the chirotope of $widehat P$, then there exists a direct affine transform that maps each point of $Q$ within distance $varepsilon$ of the corresponding point of $P$.