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# Convex Mirror (Principal Axis) watch

1. (I'm working in 2 dimensions here.)

Say we have a circular mirror with radius R and length 2R*Phi. We take an object and place it some point and get a real image at point i.

Now say we shorten the mirror, but only on 1 side of the principal axis. (Obviously the whole mirror is shortened as a result, I just want to ensure that the principal axis changes.) My intuition suggests that the image will remain at the same point i, because the major rays and their angles of reflection - and therefore the point i of their intersection - hasn't changed. But on the other hand, the principal axis does change, suggesting the location of the image will change.

For example, I can imagine changing the orientation of the principal axis so that the projection of the point p onto the new principal axis now falls on the new focal point, in which case no image is formed at all.

Can someone help me resolve this seeming paradox?

Thanks,
Eden
2. (Original post by eds_pds)
(I'm working in 2 dimensions here.)

Say we have a circular mirror with radius R and length 2R*Phi. We take an object and place it some point and get a real image at point i.

Now say we shorten the mirror, but only on 1 side of the principal axis. (Obviously the whole mirror is shortened as a result, I just want to ensure that the principal axis changes.) My intuition suggests that the image will remain at the same point i, because the major rays and their angles of reflection - and therefore the point i of their intersection - hasn't changed. But on the other hand, the principal axis does change, suggesting the location of the image will change.

For example, I can imagine changing the orientation of the principal axis so that the projection of the point p onto the new principal axis now falls on the new focal point, in which case no image is formed at all.

Can someone help me resolve this seeming paradox?

Thanks,
Eden
It's an interesting puzzle. I take it by "shorten" the mirror you mean make it smaller by removing, say, a part of it at the top?
The intuitive answer is correct. The image will not change size or position.
Ray diagrams are mathematical constructions to explain reality. In this case, you can either leave the principal axis where it was (even though it should no longer be called that) and the ray diagram will be the same.
If you rotate the principal axis so that it now passes through the new pole of the mirror (rotate about the centre of curvature) the position of the object will now be different w.r.t. the mirror and axes. Drawing a new ray diagram, with new object position and orientation, will still, however, produce an image in the same place it was before.

The 2nd diagram shows the effect of rotating the principal axis to account for, for example, the fact that you have removed part of the top of the mirror. If you draw the principal rays you can see that the tip of the image arrow falls in the same place in space, though a different place wrt the geometry.
I can assure you that if you do the same for the bottom of the object arrow, its image point will also be in the same place in space that it was before.

3. That's exactly what I meant, Stonebridge. Excellent answer. Thank you very, very much!

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Updated: March 18, 2011
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