Finding Reflection Points within a Rectangle

Date: 03/24/2005 at 23:39:47
From: Bill
Subject: Reflections within a rectangle

Given a beginning point and an ending point inside a rectangle, I'm
trying find a formula, algorithum or calculation that would tell me 
where on the rectangle I would have to aim a laser from the given 
beginning point, to a first contact point (not given) on the rectangle 
so that it would reflect and continue to a second point (not given) on 
the rectangle and reflect again so that it connects to the given 
ending point. 

The rectangle is 40 inches by 80 inches.  Basically the laser has to 
contact exactly two sides of the rectangle before contacting the end 
point. 

I've tried using triangles and parallelograms but I'm just not 
finding the solution.

Date: 04/04/2005 at 13:38:36
From: Doctor Douglas
Subject: Re: Reflections within a rectangle

Hi Bill.

There are of course multiple answers:  if the starting point is A
and the ending point is B, and you label the four walls by their
cardinal directions (N,E,S,W), then you can probably intuitively
generate solutions such as A-E-W-B and A-W-E-B.  My guess is that
you want the SHORTEST distance solution of the various choices 
that satisfy the two-bounce condition.

Here's what I suggest.  Draw a grid of rectangles that include
the original rectangle, and in each construct the image of "B" by
reflection from each wall.  You will have to construct the
coordinates for each image, but I don't think you will have much
difficulty in doing that.  In the diagram below, PQRS is the
rectangle, with given start A and end B.  Each reflected image of B in
the other rectangles is represented by a numeral:

   +---------+---------+---------+---------+---------+
   |       4 | 3       |       2 | 3       |       4 |
   |         |         |         |         |         |   View this
   +---------+---------+---------+---------+---------+   diagram in
   |         |         |         |         |         |   a monospaced
   |       3 | 2       |       1 | 2       |       3 |   font.
   +---------+---------P---------Q---------+---------+
   |       2 | 1       |       B | 1       |       2 |
   |         |         |A        |         |         |
   +---------+---------S---------R---------+---------+
   |         |         |         |         |         |
   |       3 | 2       |       1 | 2       |       3 |
   +---------+---------+---------+---------+---------+
   |       4 | 3       |       2 | 3       |       4 |
   |         |         |         |         |         |
   +---------+---------+---------+---------+---------+

There is one image (B) for which a direct path requires no 
reflections.  The four cells immediately adjacent (N/E/S/W) to the 
central box require one (and only one) reflection, so that they (1's) 
can be ignored for your application.  The next set of surrounding 
boxes require exactly two reflections (2's).  It is from these eight
choices of "2" that you have to select the one nearest to the start
point A.  In the above diagram the northwest and southwest choices are
the two logical candidates for the closest point.

If you draw a line from A to the closest image from among the eight
choices, you will see that it crosses exactly two sides.  These
crossings represent the needed reflection points.  

- Doctor Douglas, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 04/06/2005 at 21:14:54
From: Bill
Subject: Re: Reflections within a rectangle

I wanted to say thanks, I appreciate your help.  Your sample was very
easy to follow. 

Sincerely,

Bill