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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