Symmetries of quadratic forms classes and of quadratic surds continued fractions. Part I: A Poincar\'e model for the de Sitter world

The problem of the classification of the indefinite binary quadratic forms with integer coefficients is solved introducing a special partition of the de Sitter world, where the coefficients of the forms lie, into separate domains. Every class of indefinite forms, under the action of the special linear group acting on the integer plane lattice, has a finite and well defined number of representatives inside each one of such domains. This property belongs exclusively to rational points on the one-sheeted hyperboloid. In the second part we will show how to obtain the symmetry type of a class as well as its number of points in all domains from a sole representative of that class.