Beam Detector

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

A ``beam detector'' for a given curve $C$ is defined as a curve (or set of curves) through which every Line tangent to or intersecting $C$ passes. The shortest 1-arc beam detector, illustrated in the upper left figure, has length $L_1=\pi+2$. The shortest known 2-arc beam detector, illustrated in the right figure, has angles

$$\theta_1 \approx 1.286{\rm\ rad}$$ (1)

$$\theta_2 \approx 1.191{\rm\ rad},$$ (2)

given by solving the simultaneous equations

$$2\cos\theta_1-\sin({\textstyle{1\over 2}}\theta_2)=0$$ (3)

$$\tan({\textstyle{1\over 2}}\theta_1)\cos({\textstyle{1\over ... ...over 2}}\theta_2)[\sec^2({\textstyle{1\over 2}}\theta_2)+1]=2.$$ (4)

The corresponding length is

$L_2=2\pi-2\theta_1-\theta_2+2\tan({\textstyle{1\over 2}}\theta_1)+\sec({\textstyle{1\over 2}}\theta_2)$
$-\cos({\textstyle{1\over 2}}\theta_2)+\tan({\textstyle{1\over 2}}\theta_1)\sin({\textstyle{1\over 2}}\theta_2)=4.8189264563\ldots.\quad$	(5)

A more complicated expression gives the shortest known 3-arc length $L_3=4.799891547\ldots$. Finch defines

$$L=\inf_{n\geq 1} L_n$$ (6)

as the beam detection constant, or the Trench Diggers' Constant. It is known that $L\geq\pi$.

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.