A method and apparatus for providing low-cost, realistic force feedback including an improved actuator. The invention provides force sensations to a user and includes an interface device coupled to a host computer and allowing a user to interact with a host application program. A user object, such as a joystick, is moveable by a user in at least one rotary degree of freedom. A sensor reports a locative signal to the host computer to indicate a position of the user object. An actuator outputs forces on the user object in response to signals from the host computer and program. The actuator includes a housing, a set of grounded magnets provided on opposing surfaces of the housing and creating a magnetic field, and a rotor coupled to the user object positioned between the magnets. The rotor rotates about an axis of rotation and includes a shaft and teeth spaced around the shaft. An electric current flows through one or more coils on the teeth to cause the rotor to rotate. The teeth and the magnets are provided in a skewed, helical arrangement relative to each other so that, as the rotor rotates, a first tooth gradually exits the magnetic field as the next consecutive tooth gradually enters the magnetic field, thereby significantly reducing a cogging effect of the rotor when the user object is moved by the user and increasing the fidelity of forces experienced by the user.

 
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> Computational geometry using control geometry having at least two dimensions

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