Az Eszterházy Károly Tanárképző Főiskola Tudományos Közleményei. 1994. Sectio Mathematicae. (Acta Academiae Paedagogicae Agriensis : Nova series ; Tom. 22)

SZILASSI, L., A computer-aided.démonstration of the Poincare model of hyperbolic geometry

A com pul iT-ai<lr<l (Icinoii.st rat ion of 135 Take the points A and H on the half line with starting point 0. Let a — d((TÄ), b = d(ÖB), and p the radius of the circle .s perpendicular to [OB) (and to Á ) and passing through IL Further, let v,(.l) = C and c = d{0C). (C is the image of the point A under the reflection with respect to the line .s on the hyperbolic plane.) From ^,

s(.l) = C it follows that p

2 = (p + b - a)(p + b — c). Also, .s J_ k, whence (*> + P

Y P Thus c = 2 br •a(r +b ) . Thus applying the sequence of inversion mentioned •

2-)-6

2 —2ab above, we obtain the following recursive formula for the sequence a

n : «o = 0, (il = k (where h < r is an arbitary real number) , 2a

n _ i -r

2 —a

n _2 •(''"' Cln = that lim a „ = /•. -2a, if n > 2 is an integer. It can be shown Thus far we can only construct points corresponding to natural numbers on those number lines of the hyperbolic plane, which appear as diameters on the P-model. The distance between the hyperbolic points A and H (the hyperbolic measure of the segment AB) is obtained by h{AB) — | |1XL(6' V AB)\ on the Cayley-Klein model (where collinear points appear as collinear points), where c is au arbitrary constant, [ and V are the end points of the chord (or diameter) containing 1 and 11. Transformations between the two models fix points of the diameters of the (common) circle of inversion, so this formula can be used also in this case, if the constant c is chosen so that whenever also holds k = «i h(OE

x) = 1 As (UVOEx) = ^ uo EI V r±k_ r-k < 1 from the équation = f \]n(UVO E\ )| we have 1 = c • In thus c = So for any * \/ r-- lr