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Suppose you have a bundle of light rays with origin around a point $x_0 \pm \epsilon_x$ and direction of about $v_0 \pm \epsilon_v$ .Is there a pair of mirrors separated by a gap like in the image such that the bundle could be trapped bouncing back and forth between the mirrors indefinitely? enter image description here

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A two sheet hyperboloid of revolution reflects any light ray directed towards one focus, so that the reflection heads towards the other focus. Note that the reflection then dies not go beyond the center of the hyperboloid. If you start out between the sheets and headed towards one focus, the reflections bounce back and forth between the sheets and never gets past the center nor cones back out. Draw it out in two dimensions (which a recall the light beam actually explores) and see what happens.

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