Finding Reflection Points within a RectangleDate: 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 |
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