Shewchuk 97 gives me the orientation of 4 points, by finding the sign of a determinant, where the matrix is composed of the coordinates of the points. So, the signed volume of a tetrahedron, or which side of a plane a point is on, where the plane is defined by the other 3 points.
Bernstein, Fussell 09 (section 3.1) has a predicate which tells me which side of a plane a point is on, where that point is defined implicitly by the intersection of 3 planes. (A product of two determinants, where the matrix is composed of the coefficients of the plane equations of the 4 planes.)
My question: Is it possible to construct a predicate like the Shewchuk orient3d, but expressed in terms of points defined implicitly by planes? So, the orientation of a tet, where the 4 vertices are each defined by 3 arbitrary planes.
I can use an arbitrary precision arithmetic library, but I'm still hopeful it's possible without it, as I'd like to make use of the floating-point expansions and adaptive evaluation from the Shewchuk paper for performance reasons. (It's taken me a year since my last question to stumble upon the idea of representing the point as 3 planes.)