Phase space warping (PSW) methodology reconstructs a non-stationary hidden process from quasi-stationary observable dynamics, where these two coupled dynamical processes have disparate time scales. PSW has been applied to multivariate damage identification and tracking, biomechanics, and manifold characterization in nonlinear dynamical systems. However, its theory is not clearly connected to its practice. Furthermore, there is no associated sampling theory or guidelines for optimal parameter selection to estimate the hidden dynamics reliably. This paper focuses on a geometrical interpretation of PSW that coherently bridges its theory and practice by providing the needed theoretical insights and explaining practical constraints. The corresponding algorithm's parameter space is explored to provide reliable and accurate estimates of the PSW function guided by the obtained geometrical properties and insights. Numerical examples of a nonlinear hierarchical dynamical system with various hidden processes and observable dynamics are used to guide the parameter selection for the PSW algorithm. Parameter selection guidelines are obtained through global sensitivity analysis to the estimation accuracy of the simulation results. The established guidelines are used to extract fatigue damage evolution in 3D-printed beams from experimentally obtained vibration data. The obtained results show how the PSW-based fatigue tracking can be used for early fatigue damage detection.