Heron's Problem: What is it?
A Mathematical Droodle
This one is a basic optimization problem. It's quite famous, being discussed in Heron's Catoptrica (On Mirrors from the Greek word Katoptron Catoptron = Mirror) that, in all likelihood, saw the light of day some 2000 years ago.
Two points A and B are given on the same side of a line l. Find a point M on l such that the sum of distances from A and B to M is minimal.
Think of l as a mirror. Heron posed that the shortest distance between A and B via l is exactly the path traversed by a ray of light emitted from A and observed at B. From here he deduced that when light is reflected in a mirror the angle of incidence fi is equal to the angle of reflection fr, where the angles at hand are formed by AM and MB with the perpendicular to l at M.
Let B' be the symmetric image of B in l, such that BB' is perpendicular to l and is divided by the latter in halves.
AM + MB' ≥ AB'.
with the equality reached only when M lies on AB', in which case clearly the two angles coincide.
Heron's problem has applications to curved surfaces, ellipse for one.
Note: the problem has been known in Japan. It appeared on a now lost sangaku tablet from 1830 written in the Yamagata prefecture [Fukagawa and Pedoe, problem 4.1.1]. The Japanese variant underscored the computational aspects of the solution:
P is a variable point in the given segment AB, and C is a fixed point on the perpendicular AC to the line AB, and D is fixed, on the same side of AB as C, and lies on the perpendicular BD to AB.
It is not difficult to surmise the solution from the diagram
When P satisfies the minimality requirement, triangles APC and BPD' are similar from which
AP / AC = BP /BD = (AP + BP) / (AC + BD),
p / c = k / (c + d),
p = kc / (c + d).
H. Fukagawa, D. Pedoe, Japanese Temple Geometry Problems, The Charles Babbage Research Center, Winnipeg, 1989
Charles Babbage Research Center
P.O. Box 272, St. Norbert Postal Station
Canada R3V 1L6
- P. J. Nahin, When Least Is Best, Princeton University Press, 2007 (Fifth printing).
- V. M. Tikhomirov, Stories about Maximua and Minima, AMS & MAA, 1990
- D. Wells, The Penguin Book of Curious and Interesting Geometry, Penguin, 1991
- I. M. Yaglom, Geometric Transformations I, MAA, 1962
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