You are in a large gallery with a large rectangular painting hung high on a wall. Its lower edge is m feet above your eye level. Its upper edge is n feet above your eye level.

How far back should you stand to maximize your viewing angle, the angle from your eyes to the top and bottom of the painting.

The angle to be maximized is atn(n/x) - atn(m/x)

Deriving:

1/(1+(n^2/x^2)) * (-n/x^2) - 1/(1+(m^2/x^2)) * (-m/x^2) = 0
n/(x^2+n^2) = m/(x^2+m^2)
n*(x^2+m^2) = m*(x^2+n^2)
(n-m)*x^2 = m*n^2 - (m^2)*n
x^2 = (m*n^2-(m^2)*n)/(n-m) = m*n

x = sqrt(m*n), the geometric mean of m and n.

 

Posted by Charlie on 2014-02-18 17:18:57