Well, the parametric equation for a helix could be
So, you could have the two equations (I think you'd have to have two cartesian equations to represent a space curve, simply because of the degrees of freedom involved):
In general, if you have a helix, the axis of which is parallel to one of the three main axes, then I'd solve that equation for the parameter, and then plug back into the other two equations. For example, take the helix
I'd solve for giving and then plug into the other two equations, yielding
If the axis is not parallel to one of the three main axes, things get a lot dicier.
Does this answer your question?