Position Coordinates vs time for perfect circular motion in any number of spatial dimensions and assuming total speed of 1?

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If something has certain initial position coordinates, and certain initial velocity coordinates, its total speed is always 1, and assuming the circle has its center at the origin or the point with a value of 0 in every spatial dimension, what would be the most general parametric equation for the position coordinates vs time of the object that applies to any number of spatial dimensions?