Wigner焦散与欧几里得平面中前向曲线的中心对称集的奇点
Singularities of the Wigner Caustic and the Centre Symmetry Set of Frontal Curves in the Euclidean Plane
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中文总结 AI 辅助
将Wigner焦散与中心对称集推广至前向曲线，给出尖点判据、降阶关系及尖点计数不等式推广，并分析奇异平行对的共振传递结果。
中文摘要 AI 辅助
受Wigner焦散在半经典相空间分析中的作用的启发，我们将其与中心对称集一起，从正则平面曲线推广到余定向前向曲线。对于角正则平行对，Wigner焦散的带符号速度为扩展带符号曲率半径之差的一半，而有限中心对称集的奇点则是其投影比的临界点。这些公式给出了普通尖点和高阶尖点的判据及显式不变量，两个构造之间的降阶关系，以及严格凸卵形经典尖点计数不等式的重数加权推广。我们还分析了包含非前向的奇异前向曲线的平行对。一个$5/2$-尖点在Wigner焦散上产生一个$5/2$-尖点，或者在带符号半径共振时产生一个$5/3$-尖点。相反的共振将中心对称集发送至无穷远。我们获得了$5/3$-尖点的相应传递结果，并给出了一个射影完备化，该完备化解决了两个带符号半径的消失分母和同时有限阶零点。
英文摘要
Motivated by the role of the Wigner caustic in semiclassical phase-space analysis, we extend it, together with the centre symmetry set, from regular planar curves to cooriented frontals. For an angularly regular parallel pair, the signed speed of the Wigner caustic is one half of the difference of the extended signed radii of curvature, while singular points of the finite centre symmetry set are the critical points of their projective ratio. These formulas yield criteria and explicit invariants for ordinary and higher cusps, an order-lowering relation between the two constructions, and a multiplicity-weighted extension of the classical cusp-count inequality for strictly convex ovals. We also analyse parallel pairs containing a singular frontal which is not a front. A $5/2$-cusp produces either a $5/2$- or, at a signed-radius resonance, a $5/3$-cusp on the Wigner caustic. The opposite resonance sends the centre symmetry set to infinity. We obtain the corresponding transfer results for a $5/3$-cusp and give a projective completion which resolves vanishing denominators and simultaneous finite-order zeros of the two signed radii.
发表机构
- Warsaw University of Technology（华沙理工大学）
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