<p>This paper concerns the general axisymmetric problem in plasticity in conjunction with the hypothesis of Haar and von Karman for calculating stress fields. No other restriction is imposed on the yield criterion. The stress equations comprising the yield criterion and the equilibrium equations without body forces are statically determined in the sense that there are four equations involving only the four components of stress. Therefore, the result of the present paper is independent of the plastic flow rule. It is also immaterial whether elastic strains are included. It is shown that the problem above reduces to a purely geometric problem of determining an orthogonal coordinate system whose scale factors satisfy a parametric equation. Any orthogonal net satisfying this equation determines a net of principal stress trajectories giving a solution to the stress equations. The general method applies to finding the specific equations for several widely used yield criteria. Characteristic analysis of the equations that describe the mapping between the principal line coordinate system and a cylindrical coordinate system is performed. A numerical scheme based on the method of characteristics is developed and employed for calculating the stress field near a rotational ellipsoid whose surface is traction-free.</p>