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

136 l.aJim» S/ila>si point A on a line passing through the centre O of the P-model, for which It can be shown that //( — ./ ) = -/>(.r), hence the point A can indeed be any point on a diameter of the circle of inversion, that is — r < x < r. Now we need the inverse of this function, which maps a number line of the hyperbolic plane with origin O to the Cartesian system of the P-model. The inverse of the function h(.r) is d(.r) = r • lh ix - In yj^ , and its domain is the set of real numbers, its range is the open interval ( — /•; / ). It can be shown that for any natural number n we have a

n — d(n), so that on the P-model we can construct points of the hyperbolic number line corresponding to any (not only natural) number. The question that is arising now is how to construct a system of coordinates on the hyperbolic plane similar to that of the Cartesian system of Euclidean geometry. The Cartesian system assigns bijectively a point of the Euclidean plane to every pair of real numbers. When drawing the graph of a function, we in fact draw the set of points corresponding to pairs of numbers assigned to each other by the function. In hyperbolic geometry, this is somewhat more complex. First we assign (a suitable way) a point of the plane to every pair ( r. y), then assign the point corresponding to it on the P-model (i.e. its coordinates (i'k*!Jk) i

n the Cartesian system with origin 0), and then, if we want to present it on the computer, it should be changed to the coordinates of the screen. The latter, however, is more a problem of programming than Mathematics. The Cartesian system consists of two perpendicular number lines, usually with the same unit. The bijection between the pairs (.v. y) and the points of the plane can be realised two différent ways. The first one: We determine the points I

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l and P" on the axes X and Y corresponding to the numbers x and /y, then construct the point P as the intersection of the Unes ( and / perpendicular to A and Y and passing through P' and P"

, respectively. rf(0 X ) = .r, we have h(0 X) = h(x) lu ln 3. An "orthogonal" system of coordinates on the hyperbolic plane