  This paper describes the cutting sequences of geodesic flow on the modular
surface H/PSL(2,Z) with respect to the standard fundamental domain F = {z=x+iy:
-1/2 <= x <= 1/2 and |z|>=1} of PSL(2,Z). The cutting sequence for a vertical
geodesic {\theta+it: t > 0} is related to a one-dimensional continued fraction
expansion for \theta, called the one-dimensional Minkowski geodesic continued
fraction (MGCF) expansion, which is associated to a parametrized family of
reduced bases of a family of 2-dimensional lattices. The set of cutting
sequences for all geodesics forms a two-sided shift in a symbol space {L,R,J}
which has the same set of forbidden blocks as for vertical geodesics. We show
that this shift is not a sofic shift, and that it characterizes the fundamental
domain F up to an isometry of the hyperbolic plane H. We give conversion
methods between the cutting sequence for the vertical geodesic {\theta+it: t >
0}, the MGCF expansion of \theta and the additive ordinary continued fraction
(ACF) expansion of \theta. We show that the cutting sequence and MGCF
expansions can each be computed from the other by a finite automaton, and the
ACF expansion of \theta can be computed from the cutting sequence for the
vertical geodesic \theta+it by a finite automaton. However, the cutting
sequence for a vertical geodesic cannot be computed from the ACF expansion by
any finite automaton, but there is an algorithm to compute its first l symbols
when given as input the first O(l) symbols of the ACF expansion, which takes
time O(l^2) and space O(l).
