Two mirrors are placed in the first quadrant of the xy-plane (perpendicular to the plane). The first mirror along the line y = b-x (for some b > 0) and facing the point (0,0). The second mirror along the line y = mx (where m > 0) and facing the point (b,0). A light source at point (a,0), 0 < a < b, shoots a beam of light into the first quadrant parallel to the first mirror. Find m such that when the beam is reflected exactly once by each mirror, it passes through the original light source at point (a,0).

Bractals was nice enough to solve my Bouncy Bouncy Revisited with a nice formula for bouncing a laser beam of slope a off of a mirror of slope b.  a'=(b(ab+2)-a)/(b(2a-b)+1))

The original path of the beam has slope -1 andbounces off a mirror of slope m so we have (after simplifying)

a'= (m^2 - 2m - 1)/(m^2 + 2m - 1)

Bouncing this off the mirror of slope -1 gives

a"= 1/a' = (m^2 + 2m - 1)/(m^2 - 2m - 1)

So from here its just a matter of finding the points and getting them to align.

Posted by Jer on 2006-10-10 11:36:40