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

132 l.a jo> ^/ilassi We iiitroduce the notion of congruence axioniatically, using axioms of the reflection, at the Teacher Training College. On the one hand, this is a continuation of the way geometry is taught at primary school, on the other hand it is clearer this way how absolute geometry splits into Euclidean and hyperbolic geometry depending on what axioms we accept. The P-model is also suitable for visualising the most important notions and theoreins of absolute geometry in a différent way. This way the relation between the geometry developed axioniatically and the geometry based on our ;'experience" and "intuition" can be seen more clearly. Any graph—like graphs of geometrical configurations on the P-model—can only become really suggestive if we can direct drawing and see ourselves how the graphs change while changing the parameters. This aim is ser ved by the interactive computer programme to be presented. It enables the user that he himself can draw the graphs, which are to make notions of hyperbolic geometry more suggestive. 1. General description of the programme Ln the first part of the programme we can draw some basic geometrical configurations on the P-model on the screen, e.g. a line determined by two of its points, a segment with its perpendicular bisector, a regulär curve (i.e. a circle, a paracycle, a hypercycle or a line) passing trough three points, etc. It suffices to make a procédure which draws the line of the hyperbolic plane through two points on the P-model. Using this interactively, we can easily visualise the relation of two lines. Creating such a subroutine is only a problem of calculation and programming. From the parameters of the circle of inversion k(0, r) and the points P and Q (with respect to the Cartesian system of coordinates on the screen) first we had to determine the inverse P' = <Pk(P) of the point P with respect to the circle 7,-, then the parameters of the circle s passing through P, P', Q, finally the arc of s contained in k (or, if s happens to be a line, a diameter of A:). We have chosen to use the polar coordinates (with respect to 0) of