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

134 l a jus S/ila>M In each of the three cases those regulär curves of the pencil were drawn, which intersect the Unes of the pencil in points at a constant distance from each other. The user can modify both this distance—which was chosen of unit length—and other parameters of the model, within certain hmits, in order to realise connections between parameters and the graph thus obtained. Finally we constructed a system of coordinates similar to that of the Cartesian system of the Euclidean plane, and drew the graphs of a few functions m this system. Without describing the technical détails of pro gramming, we are presenting some

—mostly mathematical

—problems, which can lead to the graphs of the figures, including the graphs of functions. First we want to see how to draw a number line of the hyperbolic plane on the P-model. Let this "line" be the diameter < of the circle k. We assign 0 to the centre 0 = Kq , 1 to an arbitrary point E\ of the hne c. The point A

2 corresponding to 2 can be constructed the following way. We draw the "line'" perpendicular to e and containing E\ — call this line /] —, then mvert the point A

0 to this "line": A

2 - ^(Aq). Similarly, A

3 = y/

2(A

1), where /

2 _L ( and G A

2 h G A 2

, and so on. This way we have obtained lines on the P- model perpendicular to a given line and intersecting it in points corresponding to integers. So we must find a sequence n

„ for which h(0E\) = 1, d(0E\) = k — a 1, h(OE

n) — n, and d(OE

n) = a

n where k < r are arbitrary real numbers. 2. Number line on the P-model Obviously, we will need the "screen-coordinates" of these points for drawing on the screen. Howe ver, we are now going to use a Cartesian (i.e. Euclidean) system of coordinates with the centre of the circle k as origin. (The transformation taking these systems into each other is a problem of programming.) In what follows, h(AB) stands for the distance of the points .4 and B 011 the hyperbolic plane, d(Afí) for their distance on the P-model (i.e. on the Cartesian system mentioned above.)