  Here I present a method how intersections of a certain density matrix of rank
two with the zero-polytope can be calculated exactly. This is a purely
geometrical procedure which thereby is applicable to obtaining the zeros of SL-
and SU-invariant entanglement measures of arbitrary polynomial degree. I
explain this method in detail for a recently unsolved problem. In particular, I
show how a three-dimensional view, namely in terms of the Boch-sphere analogy,
solves this problem immediately. To this end, I determine the zero-polytope of
the three-tangle, which is an exact result up to computer accuracy, and
calculate upper bounds to its convex roof which are below the linearized upper
bound. The zeros of the three-tangle (in this case) induced by the
zero-polytope (zero-simplex) are exact values. I apply this procedure to a
superposition of the four qubit GHZand W-state. It can however be applied to
every case one has under consideration, including an arbitrary polynomial
convex-roof measure of entanglement and for arbitrary local dimension.
