On the trapping of ray families by mirrors
Abstract
We settle two questions of Serge Tabachnikov regarding the trapping of light rays by mirrors. We show (i) that the largest dimension of a trapped family of rays in ℝn is 2n−3, and (ii) that in ℝ3 one can trap a 2-parameter family of rays that is not locally normal to any smooth surface. We first establish that no 2n−2-parameter family can be trapped by showing that no open subset of Ln can be trapped due to Poincaré's recurrence theorem, yielding an upper bound of 2n−3. We then show that a system of mirrors traps a 2n−3 parameter family of rays by noting that a compact subset of the set of states of rays in the central "waist" of the trap is a normally hyperbolic invariant manifold, making it amenable to a stable manifold theorem for NHIMs, which implies that the set of rays asymptotic to the waist of the trap has dimension 2n−3. We then truncate our mirror system, which permits the entrance of rays from arbitrarily far outside of the trap, attaining the upper bound 2n−3 and resolving the first question. We then show that in ℝ3, there exists within the trapped family of rays a 2-parameter subfamily on which the canonical symplectic form on the space of oriented lines is nowhere vanishing, implying that the subfamily is not locally normal to any smooth surface, which settles the second question.
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BibTeX
@misc{omalley2026trapping,
title = {{On the trapping of ray families by mirrors}},
author = {Casey O'Malley},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.03257},
eprint = {2609.03257},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.03257},
}
RIS
TY - UNPB
TI - On the trapping of ray families by mirrors
AU - Casey O'Malley
PY - 2026
DA - 2026/09/03
PB - arXiv
DO - 10.48550/arxiv.2609.03257
UR - https://arxiv.org/abs/2609.03257
AB - We settle two questions of Serge Tabachnikov regarding the trapping of light rays by mirrors. We show (i) that the largest dimension of a trapped family of rays in ℝ^n is 2n−3, and (ii) that in ℝ^3 one can trap a 2-parameter family of rays that is not locally normal to any smooth surface. We first establish that no 2n−2-parameter family can be trapped by showing that no open subset of L_n can be trapped due to Poincaré's recurrence theorem, yielding an upper bound of 2n−3. We then show that a system of mirrors traps a 2n−3 parameter family of rays by noting that a compact subset of the set of states of rays in the central "waist" of the trap is a normally hyperbolic invariant manifold, making it amenable to a stable manifold theorem for NHIMs, which implies that the set of rays asymptotic to the waist of the trap has dimension 2n−3. We then truncate our mirror system, which permits the entrance of rays from arbitrarily far outside of the trap, attaining the upper bound 2n−3 and resolving the first question. We then show that in ℝ^3, there exists within the trapped family of rays a 2-parameter subfamily on which the canonical symplectic form on the space of oriented lines is nowhere vanishing, implying that the subfamily is not locally normal to any smooth surface, which settles the second question.
ER -