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Kustaanheimo-Stiefel (KS) transformation depends on the choice of somepreferred direction in the Cartesian 3D space. This choice, seldom explicitlymentioned, amounts typically to the direction of the first or the thirdcoordinate axis in celestial mechanics and atomic physics, respectively. Thepresent work develops a canonical KS transformation with an arbitrary preferreddirection, indicated by what we call a defining vector. Using a mix of vectorand quaternion algebra, we formulate the transformation in a reference frameindependent manner. The link between the oscillator and Keplerian firstintegrals is given. As an example of the present formulation, the Keplerianmotion in a rotating frame is re-investigated.

Kustaanheimo-Stiefel transformation with an arbitrary defining vector