仿射反射变换的不变量
Invariants of an Affine Reflection Transform
- The University of Liverpool（利物浦大学）
- Warsaw University of Technology（华沙理工大学）
机构由 AI 辅助整理，请以论文原文为准。
中文总结 AI 辅助
本文提出仿射反射变换（ART）并研究其奇点、离散化、面积性质及仿射不变度量，证明尖点数量下界、收敛性及能量上界等不变量结果。
中文摘要 AI 辅助
我们通过将平行切线对合应用于通过内部点 p 的弦并取其包络，引入了卵形线 C 关于内部点 p 的仿射反射变换（ART）。一种对偶射影方法给出了无坐标的奇异性判据，并证明了每个一般的非退化 ART 具有奇数个普通尖点，至少三个。它与中心对称集和 Wigner 焦散的关系产生了进一步的尖点界限。\n我们为具有平行对边的凸多边形定义了离散 ART。其组合单元允许显式有理参数化。非退化单元是椭圆或双曲线的弧，绝不是抛物线。在细化切线逼近下，离散变换在 Hausdorff 度量下收敛到光滑变换。\n光滑和多边形变换的有向面积是非正的，并允许 Sobolev 型解释。最大化其绝对值产生一个仿射不变的不对称性度量和集值仿射中心。我们建立了边界退化和消失结果，刻画了中心对称性，并证明了最大化轨迹不必位于中心对称集或 Wigner 焦散上。最后，我们证明了归一化能量的普遍上界，并加强了常宽卵形线的结果。
英文摘要
We introduce the affine reflection transform (ART) of an oval $C$ relative to an interior point $p$ by applying the parallel-tangent involution to chords through $p$ and taking the envelope. A dual-projective approach gives a coordinate-free singularity criterion and proves that every generic non-degenerate ART has an odd number of ordinary cusps, at least three. Its relation to the centre symmetry set and the Wigner caustic yields further cusp bounds. We define a discrete ART for convex polygons with parallel opposite sides. Its combinatorial cells admit explicit rational parametrisations. Non-degenerate cells are arcs of ellipses or hyperbolas, and never parabolas. Under refining tangent approximations, the discrete transform converges in the Hausdorff metric to the smooth one. The oriented area of the smooth and polygonal transforms is non-positive and admits a Sobolev-type interpretation. Maximising its absolute value produces an affine-invariant asymmetry measure and a set-valued affine centre. We establish boundary degeneration and vanishing results, characterise central symmetry, and show that the maximising locus need not lie on the centre symmetry set or Wigner caustic. Finally, we prove universal upper bounds for the normalised energy, strengthen them for ovals of constant width.
