We consider here a generalization of a well known discrete dynamical system produced by the bisection of reflection angles that are constructed recursively between two lines in the Euclidean plane. It is shown that similar properties of such systems are observed when the plane is replaced by a regular surface in ℝ3 and lines are replaced by geodesics. An application of our results to the classification of points on the surface as elliptic, hyperbolic or parabolic is also presented.