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 computer-aided démonstration of the Poincare model of hyperbolic geometry LAJOS SZILASSI Abstract. Teaching non-Kuclidean geometries for students al the Juhász ( ivula leachers lraining College, seems to be an effective way of developing their visitai imagination. This is also an objective of well-ktîown Kuclidean models of hyperbolic geometry. In particular we novv deal with the circle-model of Poincare we have thought to be more suggestive because of its property of preserving angles. At this lecture we are presenting a computer programme which besicles illustraling the basic notions of hyperbolic geometry (hyper cycle, paracvcle. penciles of littes etc.) also démont rates the problern described above. It présents a a "('artesian-like" system of co-ordinat.es on the Poincare model. on which the graphs of soute well-know functions can be studied in this systern of co-ordinates. We can see that the smaller the; unit is chosen comparittg to the radius of the basic circle the more the grpah approaches its usual graph. This is an effective way to make prospective teachers aware that when in the school they say '"The graph of the function Y ~ X is a line", they virtually state an équivalent form of the Euclid's parallel axioni. At lectures in Geometry at the Juhasz Gyula Teacher Training College, teaching non-Euclidean geometries is an effective way of developing a visual approach to Geometry in students. For prospective teachers, it is especially important that these topics, which require a higher level of abstraction, be treated a visual and suggestive way, as this is how they will be able to make the most use of their studies when teaching. This aim is also served by Euclidean models of hyperbolic geometry. We are now considering the Poincare model (P-model in what follows), which we have found more suggestive than other models, due to its property of preserving angles. The reason why this model is treated less frequently is probably that figures seem to be somewhat more fastidious to draw than, for instance, in the Cayley-Klein model, which uses methods of projective geometry. Using computer, however, it is not much more difficult to draw an arc, for example, than a segment of line. The analogy between the axial reflection of the Euclidean plane and the inversion on the P-model is also more suggestive than the central collineation on the Cayley-Klein model.