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The problem of the classification of the indefinite binary quadratic formswith integer coefficients is solved introducing a special partition of the deSitter world, where the coefficients of the forms lie, into separate domains.Every class of indefinite forms, under the action of the special linear groupacting on the integer plane lattice, has a finite and well defined number ofrepresentatives inside each one of such domains. This property belongsexclusively to rational points on the one-sheeted hyperboloid. In the secondpart we will show how to obtain the symmetry type of a class as well as itsnumber of points in all domains from a sole representative of that class.

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