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It is known that NOT all triangles on the hyperbolic plane have a circle that contains the triangle and passes thru all its 3 vertices. IOW, the circumcircle is not a universal property of triangles.

What about the incircle? If every triangle does have a unique largest circle contained within it, how do we characterize this circle - for example, will the center of the incircle always be the intersection of angle bisectors?

Blue
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Nandakumar R
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1 Answers1

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Yes, in hyperbolic geometry all triangles have an incircle, whose center lies on all three angle bisectors of the triangle. The radius of this incircle is always at most $\tanh^{-1} (1/2) \approx 0.5493$.

Magma
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  • Thanks. And the upper limit on the radius looks intriguing. can you suggest a reference where these things can be learnt from the ground up? – Nandakumar R Apr 25 '22 at 16:15
  • And one wonders if in elliptic planar geometry, given any three points there may be more than one circle passing thru all of them! – Nandakumar R Apr 25 '22 at 16:17
  • The incircle radius of any is bounded by that of the largest possible triangle, the triangle with three ideal vertices. In the Klein model you can draw this ideal triangle as an equilateral triangle inscribed in the limit circle, and since the three sides appear half a euclidean unit away from the origin in model coordinates, they are $\tanh^{-1}(1/2)$ units away from the origin in hyperbolic space. – Magma Apr 25 '22 at 18:05
  • In elliptic geometry there are generally four circles passing through any three points in general position, corresponding to the eight circles you get in spherical geometry by replacing any subset of three given points by their antipodes. – Magma Apr 25 '22 at 18:06
  • I have no go-to textbook references for hyperbolic geometry, but if you really want to get a good intuitive grasp for hyperbolic geometry you could try playing the game HyperRogue for a while. – Magma Apr 25 '22 at 18:06