Trapping the light of a point source with segment mirrors
Consider finitely many two-sided mirrors in the plane, each a line segment, pairwise disjoint as closed segments, with reflection taking place on the open segments (endpoints excluded). A point source lies off all the mirrors. A light ray from the source is trapped if it never reaches a point strictly outside the convex hull of the mirrors.
O'Rourke and Petrovici conjectured that no such collection traps all the light from a source, along with stronger conjectures implying this. The strongest, that no collection supports a trapped nonperiodic ray, was disproved by Stephens with eight mirrors; Mitchell, Simon and Zhao showed that any finite number of rays can be trapped nonperiodically. Milovich proved that when the angles between the mirrors' lines are rational multiples of $\pi$, all but countably many rays escape.
Problem. Is there a finite collection of disjoint two-sided segment mirrors and a source point off the mirrors such that every light ray from the source is trapped?
Source: O'Rourke and Petrovici (2001), Conjecture 9; Problem 31 of The Open Problems Project (Demaine, Mitchell and O'Rourke).
Imported from topp.openproblem.net.
Classification
- Primary
- MG — Metric Geometry
- Secondary
- DS
- Keywords
- light trapping, mirrors, polygonal billiards, The Open Problems Project
References
@inproceedings{orourke2001,
author = {O'Rourke, Joseph and Petrovici, Octavia},
title = {Narrowing Light Rays with Mirrors},
booktitle = {Proc. 13th Canad. Conf. Comput. Geom.},
year = {2001},
pages = {137--140},
}
@article{mitchell2012,
author = {Mitchell, Zachary and Simon, Gregory and Zhao, Xueying},
title = {Trapping light rays aperiodically with mirrors},
journal = {Involve, A Journal of Mathematics},
volume = {5},
number = {1},
year = {2012},
pages = {9--14},
doi = {10.2140/involve.2012.5.9},
}
@article{milovich2004,
author = {Milovich, David},
title = {Trapping Light with Mirrors},
journal = {MIT Undergrad. J. Math.},
volume = {6},
year = {2004},
pages = {153--180},
}
Citing this problem
This identifier is permanent. It will not change if the problem is solved, reclassified or withdrawn.
OP-01082, "Trapping the light of a point source with segment mirrors", clopen.solutions.