The present paper proposes a new "oblique" type of elliptical vibration cutting, while the previous method is distinguished as an "orthogonal" type. The oblique type of elliptical vibration cutting is technically similar to the orthogonal type proposed previously by the present authors, but its machining principle and features are different from the previous orthogonal type. A simple analytical model of the oblique type of elliptical vibration cutting process is developed to understand its mechanics by employing the simple maximum shear stress principle. Fundamental cutting experiments are carried out by the oblique and orthogonal types of elliptical vibration cutting and the ordinary cutting, and it is confirmed that the simulation and the experiment are in a good agreement. The results show that the chip thickness, cutting force and burrs are reduced by the oblique type of elliptical vibration cutting as much as the orthogonal type, and that the oblique type is advantageous to obtain good surface finish at high vibration amplitude compared with the previous orthogonal type as the roughness due to the vibration is reduced by the obliquity.