This paper presents analytical and numerical conformal mapping (CM) to analyze magnetic fields originating from permanent magnets and armature winding currents in a slotted air-gap of a surface mounted radial permanent-magnet synchronous motor (SPM), taking into account the effect of arbitrarily curved motor surfaces. We also studied the slotless configuration of the external rotor permanent-magnet motor for the purpose of calculating the complex relative air-gap permeance. The model is able to predict the air gap field and torque with high accuracy, which cannot be achieved using the previously available analytical methods in both cases: internal and external rotor permanent-magnet motor. We obtained one of the conformal mappings used in this study, the numerical solution of the Schwarz-Christoffel (SC) integral, by using the Matlab SC Toolbox. We used this field solution to calculate cogging torque and electromagnetic torque by integrating the Maxwell stress tensor inside the air gap. We verified the accuracy of the developed method by comparing its results with those obtained from the developed numerical finite-element (FE) model.