Symbolic perturbation of plane-based geometry
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I know it's possible to apply a symbolic perturbation scheme like 'Simulation of Simplicity' to geometric predicates like the 4-point orient, to avoid handling degenerate cases. I'm assuming it's also valid to do the same with plane-based geometry, where points are implicitly defined by the intersection of 3 planes, so I can have a similar orient predicate that tells me on which side of a 4th plane the point defined by the first 3 lies. I'd perturb the coefficients of the plane equation instead of the cartesian coordinates of a point.

The problem is that a point _could_ be defined by many different planes. Each vertex in a cube  is defined by 3 planes, but the apex of a pyramid has 4. Consistency seems to be everything with schemes like SoS, and I can't figure if it matters which 3 planes I select to define a point. Perhaps it doesn't, as long as every time I refer to that point I use the same 3 planes.

So, the question: Can I choose any 3 planes to represent a point? 
