*N.B. A detailed on-line essay by S. Finch
was the starting point for this entry.*

A ``beam detector'' for a given curve is defined as a curve (or set of curves) through which every Line
tangent to or intersecting passes. The shortest 1-arc beam detector, illustrated in the upper left figure, has length
. The shortest known 2-arc beam detector, illustrated in the right figure, has angles

(1) | |||

(2) |

given by solving the simultaneous equations

(3) |

(4) |

(5) |

(6) |

**References**

Croft, H. T.; Falconer, K. J.; and Guy, R. K. §A30 in *Unsolved Problems in Geometry.* New York: Springer-Verlag, 1991.

Faber, V.; Mycielski, J.; and Pedersen, P. ``On the Shortest Curve which Meets All Lines which Meet a Circle.''
*Ann. Polon. Math.* **44**, 249-266, 1984.

Faber, V. and Mycielski, J. ``The Shortest Curve that Meets All Lines that Meet a Convex Body.'' *Amer. Math. Monthly* **93**,
796-801, 1986.

Finch, S. ``Favorite Mathematical Constants.'' http://www.mathsoft.com/asolve/constant/beam/beam.html

Makai, E. ``On a Dual of Tarski's Plank Problem.'' In *Diskrete Geometrie.* 2 Kolloq., Inst. Math. Univ. Salzburg, 127-132, 1980.

Stewart, I. ``The Great Drain Robbery.'' *Sci. Amer.,* 206-207, 106, and 125, Sept. 1995, Dec. 1995, and Feb. 1996.

© 1996-9

1999-05-26