Re: Ray inside a cone.

<pierre.bornsztein@xxxxxxxxx> wrote in message
news:1163438209.877821.14380@xxxxxxxxxxxxxxxxxxxxxxxxxxxxxxx

Hi,

Can anyone provide an elementary solution to this olympiad problem
(ussr I think) :

''A ray of light is issued from a point interior to a given cone. All
the internal surface of this cone can reflect the light.
Prove that after a finite number of reflections, the ray will never
meet the cone again.''

I have seen something dealing with symetrisation with respect to
tangent planes, but I did not understand it...

Without loss of generality, the problem can be reduced to 2 dimensions.
Consider the reflective "cone" y=|x|.

Also without loss of generality, let the ray of light emanate from the point
(y_0,0), y_0 > 0 (on the y axis) and hit the right side of the cone. If the
slope of the ray is m > -1, obviously the ray will escape the cone.

So assume the slope of the first incident ray is m <= -1.

If the slope of the emitted ray is m = -1, the ray will bounce back on the
right side of the "cone" and reverse itself still having slope m = -1,
therefore it will again escape in the opposite direction, parallel to the
"cone" side y = - x.

Now assume that the slope of the ray is m < -1. Then the first reflected ray's
slope will be m' = 1/m. This ray will be reflected once more on the left side
of the "cone". The second reflected ray will have slope m'' = -m.

But by assumption m < -1, so m'' = -m > 1, so the ray will escape again.

Now try to generalize for an arbitrary cone y = k*|x|, k > 0. This will
require finding a relationship between two slopes, that of the incident ray
and that of the reflected ray, such that the corresponding rays are symmetric
with respect to the lines y = k*x and y = -k*x.

Thanks in advance,

Pierre.
--
Ioannis
-------
The best way to predict reality, is to know exactly what you DON'T want.

.

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