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Beam Detector

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 given by solving the simultaneous equations

 (1)
 (2)

These can be found analytically as

 (3) (4)

where and are given by

 (5) (6)

with denoting the th root of the polynomial is the ordering of the Wolfram Language. The corresponding length is

 (7) (8)

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

 (9)

as the beam detection constant, or the trench diggers' constant. It is known that .

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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. 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.

Beam Detector

Cite this as:

Weisstein, Eric W. "Beam Detector." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/BeamDetector.html