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How to trap the prisoners?
By Xiaofeng Xu
This is a story I have recently read from a book which draws my great interest. It is said that once there was a cave an emperor or used to trap the prisoners. The prisoners tried to find a way to escape but the guards were somehow always one step ahead of them. The prisoners, paranoid as ever, tried to seek out the traitor who had leaked out their escape plans, only to find out that it was due to the
shape of the cave. How does this happen? Because the shape of the cave is an ellipse, the prisoners are just at one focus of the ellipse and the guards are located at the other focus of the ellipse. Therefore, the sound of the prisoners will just be heard by the guards after reflection.
Can this be proved by vigorous mathematics proof ? The answer is ‘yes’. The rule I get from the story is that after reflection, the ray passing through one focus of an ellipse will then pass through the other focus of the ellipse. This can be shown by the diagram below. Proof:
Firstly, after transferring or rotating, any ellip coordinate system with its two foci on the x
se can be shown in the rectangular -axis. Therefore, by proving that the
2
2
x y
2 A B + = ( A is greater than B ) obeys the 2 1
ellipse with the function
previous
rule I have mentioned, I can say that this is right for any single ellipse. Here is the graph for the model. The line coming from the focus F the other focus F 2 . The normal line is drawn perpendicular to the tangent through the point of contact. The incident angle is marked as l is the tangent to the ellipse. The ray is 1 . At the point of contact T it reflects and passes through α , and the reflective angle as
β .
(Picture drawn by Geometer’s
Sketchpad)
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