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In a few places I've seen talking about the upper half plane as a model of hyperbolic geometry, it's mentioned offhand that circles (in the sense of a set of points equidistant from a given point under a given metric) in the Euclidean and hyperbolic senses coincide - but that the centers are different! Is there a good way to show this?

I've tried using the hyperbolic metric by brute computation, and tried using some properties of Mobius transformations, but am stuck.

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One way to show this is to go through the Poincaré disk model. In particular:

  • The hyperbolic circles in the Poincaré disk model centered at the center point of the disk are clearly the same as the Euclidean circles centered at that point.

  • Since the isometries of the Poincaré disk model are Möbius transformations, and since Möbius transformations map circles to circles, it follows that the hyperbolic circles centered at any point in the Poincaré disk are Euclidean circles (possibly with a different center).

  • Finally, since there is a Möbius transformation that maps the Poincaré disk model isometrically to the upper half plane model, it follows that the same holds for the upper half plane model.

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