Beam Detector


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 given by solving the simultaneous equations


These can be found analytically as


where r_1 and r_2 are given by


with (P(x))_n denoting the nth root of the polynomial P(x) is the ordering of the Wolfram Language. The corresponding length is


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


as the beam detection constant, or the trench diggers' constant. It is known that L>=pi.

Explore with Wolfram|Alpha


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. R. "Beam Detection Constant." §8.11 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 515-519, 2003.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. 273, 206-207, Sep. 1995.Stewart, I. Sci. Amer. 273, 106, Dec. 1995.Stewart, I. Sci. Amer. 274, 125, Feb. 1996.

Referenced on Wolfram|Alpha

Beam Detector

Cite this as:

Weisstein, Eric W. "Beam Detector." From MathWorld--A Wolfram Web Resource.

Subject classifications