  The quantum steering ellipsoid of a two-qubit state is the set of Bloch
vectors that Bob can collapse Alice's qubit to, considering all possible
measurements on his qubit. We provide an elementary construction of the
ellipsoid for arbitrary states, calculate its volume, and explain how this
geometric representation can be made faithful. The representation provides a
range of new results, and uncovers new features, such as the existence of
"incomplete steering" in separable states. We show that entanglement can be
analysed in terms of three geometric features of the ellipsoid, and prove that
a state is separable if and only if it obeys a "nested tetrahedron" condition.
