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Consider a space with hyperbolic geometry (instead of Euclidean). We know that two objects are similar iff they are congruent. It is said that you can't capture a personal $3\times 4$ picture of yourself in such spaces:).

I am interested to know if we can develop a theory like that of convex and concave mirrors, in a hyperbolic space?

Well concave mirrors zoom in and convex zoom out.

What I'm trying to ask is: Are all mirrors in a hyperbolic space should be flat or we have other kind of mirrors?

I even don't know if we have a flat mirror in a hyperbolic spaces.

Thanks.

Qurultay
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    There are many types of mirrors, but your emphasis is not right. You need to start with the notion of a hyperbolic reflection: naturally it should be a hyperbolic isometry fixing every point of some geodesic. After that you can vary the shape of the mirror but avoiding that it is a geodesic. This will give you precisely what you want (convex and concave will correspond to those variations). – John B Mar 17 '18 at 19:26
  • But considering any shape( excluding geodesics) as mirror will give an image congruence with object, right? – Qurultay Mar 17 '18 at 19:45
  • No. Note however that the answer is also no in Euclidean geometry. – John B Mar 17 '18 at 21:21
  • I thought that all similar shapes are congruent in such spaces! – Qurultay Mar 17 '18 at 21:40
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    Regarding flat mirrors in the hyperbolic plane, they definitely exist -- check out the Hall of Mirrors in HyperRogue (direct link). Curved mirrors definitely could exist too, no problem with the law of reflection. I suppose that they would work similar to Euclidean mirrors for small distances, but more details will require some computation. – Zeno Rogue Mar 17 '18 at 23:34

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