  The renormalized trajectory in the multi-dimensional coupling parameter space
of the two-dimensional O(3) non-linear sigma model is determined numerically
under \linebreak $\delta$-function block spin transformations using two
different Monte Carlo renormalization group techniques. The renormalized
trajectory is compared with the straight line of the fixed point trajectory
(fixed point action) which leaves the asymptotically free ultraviolet fixed
point of the critical surface in the orthogonal direction. Our results show
that the renormalized trajectory breaks away from the fixed point trajectory in
a range of the correlation length around $\xi \approx 3$-$7$, flowing into the
high temperature fixed point at $\xi=0$. The analytic large $N$ calculation of
the renormalized trajectory is also presented in the coupling parameter space
of the most general bilinear Hamiltonians. The renormalized trajectory in the
large $N$ approximation exhibits a similar shape as in the $N=3$ case, with the
sharp break occurring at a smaller correlation length of $\xi \approx 2$-$3$.
