Circles, horocycles and hypercycles in a finite hyperbolic plane (Q2638517)

The author defines hyperbolic geometry HA(n) over a finite field of order \(n\equiv 3 mod 4\) in analogy to the classical Klein model. In the projective plane PG(2,n) a conic C is distinguished. Points of HA(n) are those projective points which are on no tangents (inner points), and lines of HA(n) are those lines of PG(2,n) which intersects C in two points. All classical definitions such as hyperbolic motion, reflection, rotation, orthogonality etc. carry over to this finite model. In particular circles, horocycles, and hypercycles can be defined in analogy to the classical case. The author shows that they are - together with 0, 1 or 2 absolute points (on C) - conics. To the reviewer's opinion Bachmann's work could have been also cited. The author did not discuss the case \(n\equiv 1 mod 4\).