Singularities of the Wigner Caustic and the Centre Symmetry Set of Frontal Curves in the Euclidean Plane
Year: 2026
Type: preprint
Abstract: 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.
Source: arXiv (Cornell University)
Author: Zwierzyński, Michał
Institution: Warsaw University of Technology
Language: English
Subfield: Geometry and Topology
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7213346776