Beam Detector

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

A ``beam detector'' for a given curve is defined as a curve (or set of curves) through which every Linetangent 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

			(1)
			(2)

(3)

(4)

	(5)

(6)

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.

• Gauss-Bonnet Formula
• Gauss-Bonnet Theorem
• Gauss Equations
• Gaussian Approximation Algorithm
• Gaussian Bivariate Distribution
• Gaussian Brackets
• Gaussian Coefficient
• Gaussian Coordinate System
• Gaussian Curvature
• Gaussian Differential Equation
• Gaussian Distribution
• Gaussian Distribution Linear Combination of Variates
• Gaussian Elimination
• Gaussian Function
• Gaussian Hypergeometric Series
• Gaussian Integer
• Gaussian Integral
• Gaussian Integral (Linking Number)
• Gaussian Joint Variable Theorem
• Gaussian Mountain Range
• Gaussian Multivariate Distribution
• Gaussian Polynomial
• Gaussian Prime
• Gaussian Quadrature
• Gaussian Sum