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result 1 (1) f(A,B) maps the arcs of a hyperbolic circle through A and B to the arcs of a euclidean circle through A* and B*. (2) g(A,B) maps the arcs of a euclidean circle through A* and B* to the arcs of a hyperbolic circle through A and B. (3) In the correspondence, the minor and major arcs in the hyperbolic sense correspond to the minor and major euclidean arcs.
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proof Suppose first that C lies on a hyperbolic circle K of hyperbolic radius r. By the second circumcircle theorem, C*=f(A,B)(C) lies on a euclidean circle of radius R, where 2R = sinh(r). From the definition of f(A,B), if C in in H+(A,B), then C* is in R+. If AB is a hyperbolic diameter of K, there is only one suitable arc. Otherwise, there are two arcs in R+ through A* and B* of suitable radius, one major, one minor. By the characterization of arcs, C lies on the major arc of K if and only if E(A,B,C) > 0, and C* on the major arc of the image in precisely the same cases. This identifies the image arc. The argument for C in H-(A,B) is identical.
It is now easy to see that, if C,D lie on the same arc of K, then The argument clearly reverses to give the result for g(A,B).
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