  In this paper the geometry of two and three-qubit states under local unitary
groups is discussed. We first review the one qubit geometry and its relation
with Riemannian sphere under the action of group $SU(2)$. We show that the
quaternionic stereographic projection intertwines between local unitary group
$SU(2)\otimes SU(2)$ and quaternionic M\"{o}bius transformation. The invariant
term appearing in this operation is related to concurrence measure. Yet, there
exists the same intertwining stereographic projection for much more global
group $Sp(2)$, generalizing the familiar Bloch sphere in 2-level systems.
Subsequently, we introduce octonionic stereographic projection and octonionic
conformal map (or octonionic M\"{o}bius maps) for three-qubit states and find
evidence that they may have invariant terms under local unitary operations
which shows that both maps are entanglement sensitive.
