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 .
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. Monthly93, 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.
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 
.
![tan(1/2theta_1)cos(1/2theta_2)+sin(1/2theta_2)[sec^2(1/2theta_2)+1]=2.](/images/equations/BeamDetector/NumberedEquation2.svg)
and 
are given by
denoting the 
th
root of the polynomial 
is the ordering of the Wolfram Language.
The corresponding length is
. Finch defines
.
