We consider the motion of the N-vortex points that are equally spaced along a line of latitude on a sphere with fixed pole vortices, called 'N-ring'. We are especially interested in the case when the number of the vortex points is odd. Since the eigenvalues that determine the stability of the odd N-ring are double, each of the unstable and stable manifolds corresponding to them is two-dimensional. Hence, it is generally difficult to describe the global structure of the manifolds. In this paper, based on the linear stability analysis, we propose a projection method to observe the structure of the iso-surface of the Hamiltonian, in which the orbit of the vortex points evolves. Applying the projection method to the motion of the 3-ring and 5-ring, we characterize the complex evolution of the unstable odd N-ring from the topological structure of the iso-surface of the Hamiltonian.