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Number Theory/how to determine angle of reflection, after n reflection inside a square


reflection inside a square
reflection inside a sq  
Can we determine at what angle, a line will reflect, after hitting n
times the sides of a square.

Lets say we draw a line at a certain angle on one side of a square. It
then reflect to the other side of the wall at a certain angle. And it
continues reflecting on the side of the wall of  the square n times.

Is there a general formula that can tell at what angle it will make,
after reflecting n times on the side of a square.

There is certainly a formula that can tell, given the angle it make
first with a wall and at what angle it will make on the other side of
the square, after reflecting. But what angle it will make after
reflecting n times, is there any?

Is it right, we have to go calculating for each reflection to get the
angle, after n reflection?

Isn't, this is how a computer find the angle after n reflection?

One thing is for sure, for a certain square, for each angle it make at
a certain point on one side of a square, after n reflection on the
square, it will hit a certain side at a certain point and at a certain

But I just can't think of a general formula, that can tell at what
point or at what angle it is going to hit on a certain side of the

Thank you.

If the initial angle made with the side of the square is x, and the coefficient of restitution is unity, after an odd number of bounces, the angle will be 90-x, and after an even number of bounces it will be x.
If the coefficient of restitution is e, after an odd number of bounces, n, the angle will be given by
arctan(e^n*cot(x)) and after an even number of bounces arctan(e^n*tan(x)).  This means that as time goes on, the ball gets closer to the sides of the square.



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