Transform to Canonical Position
Example: transform point 1 to (0,0,1).  Three linear equations with 8+1 unknowns.
Similarly, transform other points to: (1,0,1), (0,1,1), (1,1,1).  Get 12 equations, 4 unknowns. 
• Unique solution.
• Must be non-degenerate.  This will be true if no three points collinear.