Just as classical "Floor" continued fractions induce the Gauss map on the real unit interval, other euclidean division algorithms (Ceiling, Even, Odd, NearestInteger, ...) induce piecewise-fractional maps with integer coefficients. All these maps have natural extensions that can easily be described on an ad hoc basis. We present a general framework for dealing with all these maps in a uniform way. The construction is geometric and well visualizable, exploiting both the interpretation of continued fractions as symbolic dynamics for the geodesic flow on the modular surface, and the projective duality between the hyperbolic plane and the de Sitter space.