Invariants of an Affine Reflection Transform
Abstract
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.
Related
- Affine balancing of complex dilatations: affine invariants and optimal distortion of harmonic quasiconformal mappings
- An Affine Approach to Metric Invariants of Poncelet Polygons
- Exact ReLU realization of binary affine refinement iterates via reflection folding and cone switching
- Brown-Halmos type theorems and complex reflection groups
- An axiom of unique reflection
BibTeX
@misc{giblin2026invariants,
title = {{Invariants of an Affine Reflection Transform}},
author = {Peter Giblin and Stanisław Janeczko and Michał Zwierzyński},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27931},
eprint = {2609.27931},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27931},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Invariants of an Affine Reflection Transform
AU - Peter Giblin
AU - Stanisław Janeczko
AU - Michał Zwierzyński
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.27931
UR - https://arxiv.org/abs/2609.27931
AB - 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.
ER -